Peter Robinson told the BBC it was "possible" to reach agreement by mid-week.
Last week, Sinn Féin withdrew its support for welfare legislation.
The parties have been taking part in talks to resolve the political impasse for several days.
Deputy First Minister Martin McGuinness and Mr Robinson were due to travel to Washington for St Patrick's Day, but said on Sunday that they would stay in Northern Ireland in order to continue with negotiations.
Last week, Mr McGuinness said £200m would solve the problems over welfare reform, but on Monday, Finance Minister Simon Hamilton said no additional money could be allocated.
Mr Robinson said much was possible, but there was still a long way to go before reaching any solid deal.
"If that is possible to do, then it is also possible to have an executive meeting earlier than would normally have been be the case to look at the schemes that would be involved," he said.
"Then it would be possible to go to the speaker and the business committee and ask them to put the welfare reform bill in the assembly for next week.
"All that is possible, but it is only possible if we can reach agreements."
The first minister also said he would only consider a deal had been done on welfare when the legislation is finally passed.
Mr Robinson said: "Agreement on welfare to me now is more than getting a paper that sets out the way forward.
"It's about agreeing the schemes to go before the executive to be agreed there, and getting the welfare bill through its final stage.
"When that happens we can consider ourselves to have an agreement on welfare, but not until."
The Northern Ireland parties had agreed a deal on Westminster's welfare reform in the Stormont House Agreement in December.
The issue had previously threatened the future of the Northern Ireland power-sharing executive.
Last week, Sinn Féin withdrew support for the bill and accused the DUP of going back on what had been agreed.
The DUP rejected this, saying there had never been the money for all the things Sinn Féin wanted and that Sinn Féin had been aware of this.

Arturo Vidal scored Bayern's first when he drilled home after his initial shot had been blocked.
Germany striker Thomas Muller then turned in from close range as defender Mats Hummels nodded down from a corner.
Forward Pierre-Emerick Aubameyang and recent signing Ousmane Dembele went close for Dortmund.
"It was difficult and Dortmund played very, very well," said Ancelotti, who took over from Manchester City-bound Pep Guardiola in the summer and has won his 18th title as a coach.
"But things worked much better for us after the break and this gives us some self-confidence for the Bundesliga.
"This team is already very, very strong."
The Bundesliga season starts on Friday, 26 August, when Bayern begin their title defence against Werder Bremen.

He walked alone and in silence around the concentration and extermination camp in what was Nazi-occupied Poland where 1.1 million people, mostly Jews, were killed.
Pope Francis also met elderly survivors of the camp, kissing them on the cheeks and speaking to them softly.
He is on his third day of a visit marking 1,050 years since Poland's adoption of Christianity.
Francis has become the third Pope to walk through the main gate of Auschwitz, under its infamous inscription "Arbeit Macht Frei" - work sets you free.
But unlike his German and Polish predecessors, he is not speaking about the horrors that occurred there, says the BBC's Adam Easton in Warsaw.
Francis passed under the gates alone, wearing white robes and skullcap. After meeting the survivors, he placed a candle at the Death Wall, where prisoners were executed by the Nazis, before continuing on his own.
The Pope stopped to pray at the prison cell of Maximilian Kolbe, a Polish Catholic friar who sacrificed his life to save that of another man. The Church made him a saint in 1982.
Pope Francis knelt for many minutes in the underground cell, illuminated only by the light from a tiny window, the Associated Press news agency reports.
The Argentine Pope is on a five-day trip to Poland.
During a World Youth Day rally in the southern city of Krakow on Thursday, he urged compassion for migrants.
He told hundreds of thousands of people that "a merciful heart opens up to welcome refugees and migrants" - a statement that puts him at odds with Poland's anti-immigrant right-wing government.
The welfare reform bill could be debated in the assembly again next week if there is sufficient movement in ongoing negotiations, the first minister has said.
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Carlo Ancelotti gained his first trophy as Bayern Munich coach thanks to their 2-0 victory over Borussia Dortmund in the German Super Cup.
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The Pope has offered a private prayer at the former Auschwitz death camp.