The gist of conversations between Tony Blair and George Bush is likely to be published as part of the Chilcot inquiry into the Iraq war, after the government reached a deal in principle to make some secret information public.

The agreement between the inquiry and the Cabinet Office, which was announced on Thursday, comes after months of deadlock over what will be released, as politicians blamed each other for causing delays. Blair has repeatedly denied being responsible for the blockage, while Nick Clegg called for those likely to be criticised to accept public scrutiny and help move the process on.

The inquiry announced that agreement had been reached on the type of "gists and quotes" would be released from 25 notes and 130 records of conversations between Blair and Bush. No decision has been taken, however, on exactly which ones will be published. There is also now an agreement to release a small number of extracts from the most critical minutes of more than 200 cabinet-level discussions.

The Chilcot inquiry had requested the full content of correspondence between Blair and Bush, but appears to have watered down its demands and accepted that the gist of the conversations will be "sufficient to explain our conclusions". It may also be allowed to publish some quotes, but this will be kept to "the minimum necessary to enable the inquiry to articulate its conclusions".

The inquiry also accepts that the use of material from the letters "should not reflect President Bush's views".

Sir John Chilcot, who is leading the inquiry, has now written to the government to "record his pleasure" that agreement had been reached on an issue that had "raised difficult issues of long-standing principle".

The inquiry's website said: "Detailed consideration of gists and quotes requested by the inquiry from communications between the UK prime minister and the president of the United States has now begun. It is not yet clear how long that will take, but the inquiry and the government should work to complete the task as soon as possible."

The agreement comes after David Cameron and Nick Clegg called for the report into the Iraq war to be published as soon as possible. Cameron has said he wanted the report by the end of the year. "I think we shouldn't have to wait much longer," he said.

Clegg, who oversees the Cabinet Office, appeared to suggest last month that some of those in line for scrutiny were to blame for holding up the process, which began in 2009 and has cost millions of pounds.

"I'm sure Chilcot himself would like to get on with it as quickly as possible," Clegg said. "I can't comment on exactly the reasons why, given there's a lot of to-ing and fro-ing about what is finally produced in published form in the report.

"This involved a lot of people, a lot of legalities and, of course, is a very, very sensitive issue.

"But I really do think everybody involved, including those who know they will be subject to renewed scrutiny from the Chilcot report, all now accept that it is time that the report was published so the record can be scrutinised in the most objective way possible."

He said those who might not like being subject to scrutiny must accept that it would happen, given that it was "one of the most momentous, one of the most catastrophic decisions in British foreign policy. I would say the most catastrophic decision since Suez".

After it has been agreed exactly what will be published, the process of Maxwellisation – its name comes from a case involving Robert Maxwell – will begin, in which the inquiry puts its conclusions to those criticised.