Theresa May will discuss the state of Brexit negotiations with both the taoiseach (Irish prime minister) and DUP leader on Wednesday.
A Commons vote on Mrs May’s withdrawal plan was postponed so she could seek assurances from Brussels over the Irish border backstop.
Mrs May has been holding talks with other EU leaders.
They say the deal cannot be renegotiated.
Taoiseach Leo Varadkar has said the Irish government “will not compromise” on its commitments around the border.
Mrs May will meet Mrs Foster in London, before travelling to Dublin to speak to Mr Varadkar.
The UK is due to leave the EU on 29 March 2019.
‘Backstop needs radical surgery’
The backstop is the insurance policy arrangement aimed at avoiding the return of a hard border – physical checkpoints or infrastructure – along the 310-mile (500-km) border between Northern Ireland and the Republic of Ireland after Brexit.
The Irish government has said there can be no deal without it, and has received support from other EU leaders.
The proposal, however, has faced a backlash from Labour and other opposition parties, and the DUP – which backs the government in a confidence-and-supply pact – but says it will not back the deal unless there is “radical surgery” to amend its terms.
On Tuesday, Mrs Foster said her party was not seeking a total “rewrite” of the withdrawal deal, but that the backstop needed to be removed from the binding legal text.
If it took effect, it would see only Northern Ireland remain aligned with the EU single market in some areas, meaning new regulatory barriers between GB and NI.
The UK would also not be able to leave the backstop without EU agreement.
Mrs May is understood to be seeking legal guarantees that the UK will not be trapped in the Northern Ireland backstop plan indefinitely.
What is the Irish government’s view?
On Tuesday, Mr Varadkar said he will work to give the UK “assurances”, but never compromise on his government’s commitments around the border.
He said the Irish government also needed to up its contingency planning for a no-deal scenario in terms of recruitment of customs, veterinary and health officers at Irish ports.
“It means putting in place infrastructure at our ports and airports, in Dublin and Rosslare,” he told opposition parties in the Irish parliament (Dáil Éireann).
“Firms who don’t have action plans should develop action plans, and those who have action plans should begin to implement them.”
The UK could still withdraw the threat of no deal by revoking Article 50 or extending it, he added.
What is the EU saying?
In Brussels, European Commission President Jean-Claude Juncker said there was “no room whatsoever” for renegotiation of the deal.
He told the European Parliament the EU could give further clarification but that the withdrawal agreement, in all its parts including the backstop, would not be reopened.
Mr Juncker said the backstop was “necessary” for the entire coherence of the agreement and it was necessary for the Republic of Ireland.
“Ireland will never be left alone,” he added.
What happens next?
A summit of EU leaders is still due to take place this Thursday.
Downing Street has said a Commons vote will be held on the deal at some stage before 21 January.