Foster said talks with May would continue into next week.
He also accused Mr Adams of not being impartial, saying he was a "foreign TD from a foreign country involved in the talks".
"The issues causing this impasse are not Sinn Féin issues".
Talk of the Town understands Mr Frazer did not actually call into the Sinn Fein offices despite recording his message outside it.
Mr Adams said they would be pushing Mr Varadkar to raise the prospect of a border poll on Irish reunification within the next five years in discussions with the British government.
Northern Ireland has been without a powersharing executive since March and without a first and deputy first minister since January, after Sinn Fein collapsed the administration amid faltering trust and relations with the DUP. "They can't have it both ways, it has to be dealt with sensibly", she said.
He believed that the Stormont power-sharing arrangements would be re-established over the summer between the DUP and Sinn Fein, but with the DUP in an even stronger position than before.
"There have been positive engagements today between ourselves and Sinn Fein", he said.
"We want to see an administration set up again that will last and one that will last for all of the people of Northern Ireland", said Ms Foster outside Dublin's Government Buildings.
Speaking in Downing Street alongside Mr Varadka, Mrs May dismissed these concerns as she claimed her government "remains absolutely steadfast" in its commitment to upholding the Good Friday Agreement. "We are talking about a confidence and supply agreement with them", she said.
"Clearly the DUP are on the wrong side of the argument, cosying up alongside the Tory Government who are disrespecting the mandate of the people here, who asked to remain within the European Union", she added.
"As we spoke on the way in, I was reminded of that famous scene in Love Actually where Hugh Grant does his dance down the stairs, but apparently it wasn't filmed here so I didn't get a chance to see the stairs".