The pound hovered below a 2-1/2-year high in the Asian session following the agreement last week of a narrow Brexit trade deal that does not cover Britain's financial sector.
The dollar index was little changed at 90.37, following a three-day slide.
Sterling dipped 0.9% to $1.3432, inching away from the $1.3625 mark it hit earlier this month for the first time since May 2018.
Trump signed into law the $2.3 trillion pandemic aid and spending package, averting a partial federal government shutdown that would have started Tuesday.
Earlier he had cryptically tweeted, "Good news on Covid Relief Bill. Information to follow!" He had previously demanded an increase in stimulus checks for struggling Americans to $2,000 from $600.
The euro was little changed at $1.220.7, near the 2 1/2-year high of $1.2273 touched this month.
While last week's Brexit deal came as a relief to investors, the bare-bones nature of the pact leaves Britain far more detached from the EU, analysts say, suggesting the discount that has dogged UK assets since 2016 will not vanish soon.
Brussels has made no decision yet on whether to grant Britain access to the bloc's financial market.
Mitsuo Imaizumi, chief FX strategist at Daiwa Securities in Tokyo, expects the pound and euro to decline against the dollar, reaching $1.30 and $1.15 respectively by the end of the summer.
"Regardless of the Brexit deal, cable will be down," he said.
"It's buy the rumor, sell the fact."