By Karen Brettell

NEW YORK (Reuters) - The dollar gave up most of its early losses on Tuesday as investors waited on a Senate stimulus bill meant to blunt the economic impact of the spreading coronavirus, and after U.S. President Donald Trump pressed his case for a re-opening of the U.S. economy by mid-April.

Senior Democrats and Republicans said on Tuesday they were close to a deal on a $2 trillion coronavirus economic stimulus package, raising hopes that the divided U.S. Congress could soon act to try to limit the pandemic's economic fallout.

There is mounting frustration among some politicians that businesses are closing to stop the spread of the virus, which is leading analysts to predict the nation will see the worst recession since the Great Depression.

At the same time, many investors think markets and the economy will not be able to stabilize until the worst of the virus spread has passed.

"While we think that most currencies will eventually regain much of the ground that they have recently lost to the U.S. dollar, we don’t expect that process to start until the pandemic has clearly passed its worst," Oliver Allen, assistant economist at Capital Economics, said in a report.

Against a basket of its rivals <=USD>, the dollar fell 0.17% to 101.95. It fell as low as 101.04 on Tuesday and is down from a more than three-year high of 102.99 on Friday.

Trump has said he would like the U.S. economy to reopen by Easter, saying that Americans could continue practicing social-distancing measures, which health experts say are crucial to prevent infection, while also returning to work.

But medical experts and officials on both sides of the U.S. political spectrum say it is critical to stop the spread of the virus, whatever the economic cost.

New steps taken by the Federal Reserve on Monday to boost liquidity across debt markets and backstop lending are seen as helping market conditions.

"The Fed's measures are unprecedented, and they have been extremely proactive in preventing this external shock from morphing into a wider funding crisis," said Vasileios Gkionakis, head of FX strategy at Lombard Odier.

The U.S. central bank announced various programs, including purchases of corporate bonds, guarantees for direct loans to companies and a plan to get credit to small and medium-sized business.

Against the dollar, the euro <EUR=> was up 0.47% at $1.0771.

Some more relief was also evident in dollar funding markets, with measures of short-term funding indicators such as euro-dollar FX swaps for three-month maturities <EURCBS3M=ICAP> stabilizing around 6 basis points after blowing out to more than 100 bps last week.