DUP leader Arlene Foster has indicated a deal to support the Conservatives' minority government is close to being finalised.
The nationalist Sinn Fein and SDLP and the cross-community Alliance have all made clear Northern Ireland Secretary James Brokenshire can not chair the ongoing process to restore power-sharing at Stormont due to the perceived conflict of interest.
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".
DUP deputy leader Nigel Dodds, in Downing Street for talks with Mrs May on restoring powersharing in Northern Ireland, refused to be drawn on how the negotiations were progressing.
When pressed on whether a gay person or a woman wanting an abortion would think of the DUP as compassionate, McCausland said: "Those are matters that are devolved to the Northern Ireland Assembly".
"The form of the DUP in the past in the past, and the type of policies that they have put forward, the type of policies that they have pursued with the Tories in the past, are very very concerning", Pearse Doherty, a Sinn Fein lawmaker in the Irish parliament, said.
"I never said the British government are not honest brokers because they are putting together a deal with the DUP", he said.
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Philip Hammond has been accused of preventing Theresa May from concluding her deal with the DUP because of a row over airport taxes in Belfast.
An Irish government spokesperson said: "They discussed the outcome of the United Kingdom general election and the Prime Minister outlined the proposed supply and confidence arrangement between her party and the DUP".
But there are mounting concerns about the significant concessions that could be extracted from Mrs May over funding for Northern Ireland - potentially a billion pounds in tax breaks and infrastructure spending.
Talks prior to the election being called had failed to reach an agreement.
She said she hoped for a "successful conclusion".
A senior Tory source said talks with the DUP would continue on Wednesday.
Meanwhile, talks are continuing between the DUP and Conservatives, to secure the support of the DUP's 10 MPs in steering government business, including crucial measures on Brexit, through the Commons.
"The two issues - Northern Ireland and Brexit - might end up making the other more hard, in a vicious circle", Usherwood said.