Appointed Senator Wanchai Sornsiri, a member of the acting Senate speaker's coordinating committee, said after the meeting that most participants wanted to see such a government as soon as possible.
He said the government in question should be in place for six to 12 months in order to see through election reforms and ensure peace before elections take place.
However, he did not discuss any details of the government in question.
Sen Wanchai was speaking after the meeting led by acting Senate Speaker Surachai Liangboonlertchai and representatives from seven public and five private organisations.
The meeting had been called to discuss possible solutions to the political impasse.
The seven public organisations were the Supreme Court, the Administrative Court, the Election Commission (EC), the Office of the Ombudsman, the National Anti-Corruption Commission, the National Human Rights Commission, and the National Economic and Social Advisory Council.
The Constitutional Court and the Office of the Attorney-General did not send representatives to the meeting.
Appointed Senator Khamnoon Sitthisamarn said the EC was unlikely to proceed with the planned July 20 election if it was unsure about the status and authority of acting caretaker Prime Minister Niwatthamrong Bunsongphaisarn.
"The EC insists it will ask the Constitutional Court to convene if there are any doubts. So, the chances of having an elected prime minister look grim," he said.
Paiboon Nititawan, an appointed senator, was absent from the meeting.
Sen Paiboon said he decided to pull out because he did not believe in the approach being discussed at the meeting. He said selecting a new prime minister would be the only way to solve the political crisis.
"Holding discussions is pointless. Even so, I am looking forward to the end result," he said.
Mr Niwatthamrong said he agreed to hold talks with Sen Surachai on finding solutions to the political impasse.
However, he said he was considering asking Sen Surachai and a few other senators to attend a small group discussion to find out about their approach first.
"We need to know what the Senate's approach is and which law is being used to back it. If it turns out to be legal, the government is ready to help push for it," he said.
He also insisted he has the legal authority to seek a royal decree to endorse an election.
Pheu Thai spokesman Prompong Nopparit referred to 2008, when then deputy prime minister Chavarat Charnvirakul, acting as prime minister, submitted a royal decree for an election.
At the time his status was the same as Mr Niwatthamrong's now.
Legal expert Meechai Ruchupan said that Section 7 of the charter does not authorise the appointment of an interim prime minister. On his website meechaithailand.com, he said the law is being seen as a way to resolve unforeseen problems.