The EU has already agreed to move back the original March 29 deadline to April 12 to give May more time to win backing from MPs for the deal she negotiated with the EU, which they have rejected three times.
Labour Brexit spokesman Keir Starmer said his party wanted the talks to go on, and a spokesman for May's office said the government had "made serious proposals" in the talks and wanted them to continue over the weekend "in order to deliver a deal that is acceptable to both sides".
It is not yet known whether President Tusk will recommend they accept Mrs May's request, although he previously mooted a "flexible" extension of up to a year.
The elections pose a substantial stumbling block because Britain would be expected to take part, if it is still an EU member, so its people have representation in the European Parliament.
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In fact the June 30 date was Theresa May's request of her fellow leaders at last month's European summit.
"The important thing now is that in any extension that we get from the European Union, we have an absolute clarity that as soon as we've done the deal, we are able to bring that extension to an end", Hammond told ITV. May is the upcoming elections to the European Parliament, which begin on May 23.
Brexit is now mired in doubt, almost three years since the United Kingdom shocked the world by voting 52 percent to 48 to leave the bloc.
"The official added that Tusk went on to say: "And even if this were not possible, then the United Kingdom would still have enough time to rethink its Brexit strategy".
FILE PHOTO - German Chancellor Angela Merkel speaks at a news conference with Irish Prime Minister (Taoiseach) Leo Varadkar during Merkel's visit to Dublin, where the latest Brexit developments were on the agenda, in Farmleigh House, Dublin, Ireland April 4, 2019. She said she is making the preparations even though she believes it is not in the interest of either Britain or the European Union for her country to participate because it is leaving the bloc.
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The former Ukip leader said he would not, after 25 years of "endeavour", watch British politicians "roll us over" - as he warned of a "fightback".
Ms May said Britain would prepare for such an election, but she still hoped that an agreement would be reached sooner, allowing the extension to be ended early.
As many as 15 ministers in May's government are on the edge of resigning over her Brexit talks with the opposition Labour Party, The Sun reported.
If that doesn't work, May plans a series of votes in Parliament to see if a majority-backed strategy can emerge.
Ideas being discussed include keeping Britain in a customs union with the EU after it leaves the bloc, as well as the possibility of a second referendum.
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