Speaking alongside the President at the Elysee Palace, she said: "I confirmed to President Macron that the timetable for the Brexit negotiation remains on course and will begin next week".
Foster tweeted afterward that "discussions are going well with the government and we hope soon to be able to bring this work to a successful conclusion".
The UK Government wants the talks to take place in parallel during the Brexit process but both sides have made finding a solution to the issue of citizens' rights a priority. The government announced on Thursday, however, that the queen's speech, which outlines its legislative agenda, will go ahead next Wednesday.
Brexit negotiations that were scheduled to begin next week was also likely to be postponed.
May's botched election gamble cost her Conservative Party its majority in the 650-seat parliament last week, weakening her hand just days before talks with the European Union on a divorce that has to be agreed before an exit due in March 2019, Reuters reported.
As Mrs May attempts to cobble together a majority, the EU's Michel Barnier said he would hold talks with British envoy Oliver Robbins on Tuesday to organise the negotiations.
Mr Adams, addressing claims of Sinn Fein disengagement, said: "We want into the institutions, because that is what the people desire, that is what the people voted for".
Meanwhile, a Conservative source told Reuters talks with the DUP have been very positive and constructive so far with broad agreement on some principles, but they are not putting a time frame on when a deal will be done.
The move comes amid concerns the Government will compromise its stated impartiality in the region if it enters a confidence and supply deal with the DUP at Westminster. She reportedly apologised to Tory MPs, accepting personal responsibility for failing to win an outright victory and sacrificing the parliamentary majority she inherited from David Cameron when she became leader after the Brexit referendum previous year.
The DUP is believed to be more favorable to a "soft Brexit" that would keep Northern Ireland's border with the Republic of Ireland free-flowing.
Ms Foster said the talks deadline remained "realistic".
The Prime Minister will attend the European Council summit to discuss issues such as migration and security, but will be absent from a dinner when the 27 other national leaders review the start of the Brexit process.
"The first phase of the negotiations will tackle three main areas: safeguarding the rights of citizens, financial settlement of the UK's obligations and the new external borders of the European Union", the Commission said.