Senate Democratic Leader Chuck Schumer said his caucus has not yet made a decision on whether to withhold their votes, but added that the issue is being looked at as a way to ensure there is a thorough investigation of alleged Russian meddling in the 2016 U.S. election.
"To have that special prosecutor, people would breathe a sigh of relief because then there would be a real independent person overlooking the FBI director," Schumer told CNN's "State of the Union" program.
Trump sparked a political firestorm when he abruptly fired James Comey, director of the Federal Bureau of Investigation last week. The FBI has been investigating alleged Russian meddling in the U.S. election and possible ties between Moscow and the Trump campaign.
Russia has denied the claims and the White House says there was no collusion.
Trump, who has sought better relations with Russia, has continued to question whether Russia was behind the hacking of email accounts belonging to Democrats involved in Hillary Clinton's 2016 presidential campaign.
But Secretary of State Rex Tillerson told NBC's "Meet the Press" program on Sunday there is no question that "the Russians were playing around in our electoral processes."
He defended Trump's decision to meet with Russian Foreign Minister Sergei Lavrov in the Oval Office last week.
"It's in the interest of the American people, it's in the interest of Russia and the rest of the world that we do something to see if we cannot improve the relationship between the two greatest nuclear powers in the world," Tillerson said.
Democrats have accused Trump of attempting to thwart the FBI's probe and have called for some type of independent inquiry into the matter.
Trump has said he removed Comey because he was not doing a good job and that Comey lost the support of FBI employees.
The Justice Department began interviewing candidates for the FBI director job on Saturday. Some people under consideration include acting FBI Director Andrew McCabe, Republican Senator John Cornyn of Texas, New Appeals Court Judge Michael Garcia and former Assistant Attorney General Alice Fisher, according to a White House official.
If the Senate vote on an FBI director were to break down along strict party lines, Democrats would not have the votes to block a nominee.
Republicans control both chambers of the U.S. Congress. They hold 52 seats in the 100-member Senate, enough to approve a FBI nominee provided that no more than two Republican senators break ranks. In the event of a tie vote, Vice President Mike Pence would cast the deciding vote.
"The key is getting some of our Republican colleagues to join us," Schumer said.
Republican leaders in the Senate have rebuffed calls for a special prosecutor, saying it would interfere with ongoing congressional probes.
Republican Senator Lindsey Graham of South Carolina said on Sunday there may come a time when a special prosecutor is needed but not now.
"Right now, it is a counterintelligence investigation, not a criminal investigation. So you don't need a special prosecutor," Graham said on "Meet the Press." XAUUSD
Meanwhile, China is to release industrial production data amid recent signs of cooling in the world's second largest economy.
Elsewhere, the euro zone is to publish revised data on first-quarter economic growth as traders look for further signs on the strength of the region's economy and hints on when the European Central Bank will start withdrawing stimulus.
In the U.K., market participants will be looking ahead to reports on consumer prices, employment and retail sales for further indications on the continued effect that the Brexit decision is having on the economy.
Preliminary data on Japanese first-quarter growth will also be in focus.
Ahead of the coming week, Investing.com has compiled a list of the five biggest events on the economic calendar that are most likely to affect the markets.
1. Empire State, Philly Fed Manufacturing for May
The New York Federal Reserve Bank is to publish the Empire State manufacturing survey for May at 8:30AM ET (12:30GMT) on Monday, amid expectations for a modest improvement to 7.0 from 5.2 in April.
That is followed by the Philly Fed's own manufacturing survey due at 8:30AM ET (12:30GMT) Thursday. Market analysts expect the index to tick down to 19.8 in May from 22.0 in the preceding month.
Besides the manufacturing surveys, this week's calendar also features U.S. data on building permits, housing starts, industrial production and weekly jobless claims.
Markets are currently pricing in around a 70% chance of a rate hike in June in the wake of last week's disappointing U.S. retail sales and inflation reports, according to Investing.com's Fed Rate Monitor Tool.
Earnings from the likes of retailers such as Wal-Mart (NYSE:WMT), Target (NYSE:TGT), Home Depot (NYSE:HD), Gap (NYSE:GPS) and TJX (NYSE:TJX) are also on the radar this week, especially in light of last week's downbeat results from department store retailers like Macy's (NYSE:M) and Nordstrom (NYSE:JWN).
Headlines from Washington regarding President Donald Trump's health-care plan and tax reform will also be in focus.
2. April China Industrial Production
The China National Bureau of Statistics is to release data on April industrial production at 02:00GMT on Monday (10:00PM ET Sunday). Market analysts expect factory output to rise 7.1% last month, after increasing 7.6% in March.
The Asian nation will release fixed asset investment data and a report on retail sales at the same time.
Soft April inflation and trade data released last week underlined the view that China's economic expansion remains solid but is starting to moderate after a surprisingly strong start to the year.
3. Euro Zone Q1 GDP - Revised Reading
The euro zone will release revised first-quarter growth data at 09:00GMT (05:00AM ET) Tuesday. An initial estimate published earlier this month showed that the region's economy grew 0.5% in the three months ended March 31, accelerating from growth of 0.4% in the fourth-quarter.
Other data out of the euro zone this week include final inflation and consumer confidence figures.
In addition, market participants will be focusing on Tuesday's ZEW survey data on German economic sentiment to gauge confidence in the euro zone's largest economy.
European Central Bank President Mario Draghi reiterated last week that the ECB is in no rush to raise interest rates or wind down its massive bond-buying program, warning that it is still too early to declare victory in its quest to boost inflation despite signs the bloc's economic recovery is strengthening.
4. U.K. April CPI, Employment & Retail Sales
The U.K. Office for National Statistics will release data on consumer price inflation for April at 08:30GMT (4:30AM ET) on Tuesday. Analysts expect consumer prices to rise 2.6%, after increasing 2.3% a month earlier.
At 08:30GMT (4:30AM ET) Wednesday, the ONS will publish the monthly jobs report. The claimant count change is expected to rise by 5,000 in April, with the jobless rate holding steady at 4.7%. Wage growth including bonuses is forecast to rise 2.4%.
On Thursday, the ONS will produce a report on April retail sales at 08:30GMT (4:30AM ET), with analysts expecting an increase of 1.0%, following a drop of 1.8% in the preceding month.
Recent data has pointed to signs that rising inflation, caused in part by the pound's post-Brexit vote tumble, is crimping spending by consumers, the main drivers of the economy, just as the country is set to start EU divorce negotiations.
5. Japan Preliminary First-Quarter GDP
Japan will publish preliminary first-quarter economic growth data at 23:50GMT (7:50PM ET) on Wednesday. The report is expected to reveal that Japan's economy expanded by 0.4% in the first three months of the year, compared to growth of 0.3% in the preceding three-month period.
The economy is expected to show growth at an annualized rate of 1.7% in the first quarter, which would be the fastest growth rate since the second quarter of 2016.
Private consumption, which accounts for roughly 60% of gross domestic product, was seen rising 0.4% in the first quarter, after it showed no growth in the October-December period.
In addition to the GDP report, Japan is also slated to produce data on core machinery orders, which is looked at as a good indicator of capital spending in the coming six to nine months.