Brexit talks will begin Monday.
Conservative Party sources say May wants to show her government is up and running but her loss of authority in last week's election will make it harder to handle a hectic agenda - Brexit talks with the European Union, tackling a slowing economy, a political crisis in Ireland, and a devastating fire in London.
Ms Foster said the talks deadline remained "realistic".
"I want us to have a good working relationship so that from his jurisdiction and my jurisdiction we can work together for the good of all our people", she added.
"We would restore faith in politics if we could show that this parliament can at least function in presenting a view in the national interest which would command a majority on a cross-party basis", said veteran pro-European Conservative lawmaker Ken Clarke.
Britain's Brexit ministry said on Friday that no deal could be struck on exiting unless the future relationship with the bloc was taken into account.
In a hint at a softer approach, Mr Davis said on Monday "we will start down this process" by focusing on the divorce proceedings from the European Union before moving on to trade.
Londra, giovane coppia di italiani morti nel rogo della Grenfell Tower
Il rogo della Grenfell Tower ha provocato polemiche ma soprattutto tanto dolore: la conta dei morti non è ancora definitiva. La ditta che ha portato avanti i lavori di ristrutturazione nel 2015, però, assicura di aver rispettato tutte le norme.
Arriving to a meeting with his counterparts in the 28-country EU, Hammond said his "clear view" is "we should prioritize protecting jobs, protecting economic growth and protecting prosperity as we enter those negotiations and take them forward".
Philip Hammond, Britain's finance minister, says protecting the British economy should be the main goal of upcoming negotiations over the country's exit from the European Union.
But Mr Davis insisted the United Kingdom still wants to negotiate its future trade relationship with the European Union "alongside" talks on the terms for Brexit, which he said was the process set out in Article 50 of the European Union treaties.
Arlene Foster said the negotiations did not hinge on May's legislative program - the so-called Queen's Speech - that the British prime minister will lay out on Wednesday and Foster's Democratic Unionist Party has said it will support.
Addressing her party on Monday after one of its most memorable electoral failures, May said she would take a broader, more consultative approach to the Brexit talks.
Even with a deal nearing to secure her government's survival, Ms May is so weakened that her Brexit strategy has become the subject of public debate inside her own party, with two former Prime Ministers urging her to soften her approach.
But after a generation of discord over Europe inside her party, May's future could depend on her ability to please both the eurosceptic and pro-European factions in her party.