EU member states delayed approval of the extension on Wednesday, but European Council President Donald Tusk said earlier on Twitter he was recommending that EU leaders back Britain's request.
Johnson has said if the deadline is deferred to the end of January he would call an election by Christmas.
Foreign exchange trading was generally quiet following a sell-off in the pound on Tuesday but both the pound and euro gained against the dollar late Wednesday. Against the dollar, the pound was last up 0.37% to $1.292. The euro was up 0.1% to $1.114. Against the euro, the pound was 0.28% stronger at 86.17 pence.
The dollar index was down 0.09% at 97.436.
"After the slide in sterling we saw yesterday we are actually seeing a bit of a stabilisation this afternoon," said Shaun Osborne, chief foreign exchange strategist at Scotia Capital.
He noted, however, that "the upside for both sterling and the euro is probably fairly limited until we get some clarity on the (Brexit) situation."
Still, the pound has been able to recover "given that a no-deal Brexit has seemingly been taken off the table," said Joe Manimbo, senior market analyst at Western Union Business Solutions.
Johnson on Tuesday did not follow through on his threat to pull the deal if parliament defeated him on the timetable, which was understood by the market as all but eliminating the chances of no-deal exit.
Safe haven currencies earlier Wednesday were boosted by the Brexit uncertainty, but by late afternoon in North America, those gains had been erased. The Japanese yen was last 0.18% lower at 108.65 per dollar, with the Swiss franc at 0.990 per dollar.