Casement Park in Belfast would be 'dealt with by now' if it was football
A source last year said the meeting was at times "fraught and tetchy" and Mr Burns had accused Lyons of "dragging his heels".
Minutes obtained by BBC News NI through a Freedom of Information request were significantly redacted.
The Department for Communities (DfC) argued releasing the full documents would disclose "commercially sensitive information".
According to the minutes, Mr Burns told the minister he wanted "good relations" with the department and "candid conversations in friendship".
"He also referenced that the minister had not attended a GAA match and stressed that the invitation remains open," according to the records.
Lyons noted he had attended a GAA event the previous month and was "willing" to attend a GAA match, "but does not accept engagements on Sundays".
David Young/PA
Plans for Casement Park were developed as part of a Stormont executive funding programme agreed in 2011, which also delivered upgrades to Ravenhill rugby stadium and the football ground at Windsor Park.
Stephen McGeehan, Ulster GAA's head of operations, said the "GAA's position is that the 2011 executive commitment was to meet the strategic needs of the sporting codes".
"GAA believe that equivalent problems in other sporting codes would have been dealt with by now," the minutes added.
Lyons said that Casement Park was "treated in the same manner as the two other stadia".
He acknowledged the GAA's "frustration", but he said the Stormont executive's 2011 funding commitment "still stands".
The minister noted that "due to budget constraints, it was not possible to progress other executive capital projects, for example social housing".
But he "recognised the strategic need for sport".
Liam McBurney/PA
It was agreed, according to the minutes, that the minister would seek a meeting with the Northern Ireland Secretary, Hilary Benn, "to discuss the funding at a ministerial level".
In the Stormont assembly on Tuesday, Lyons said he had written to the secretary of state "to ask for an update on the government's view of the project".
The Northern Ireland Executive originally pledged £62.5m towards Casement Park, with the GAA providing £15m.
But the plans have suffered years of setbacks and the projected cost has risen significantly beyond the initial estimate of £77.5m.
In September, the UK government said the estimated build costs had "risen dramatically" to "potentially over £400m".
Ulster GAA secretary Brian McAvoy said this cost was "wildly exaggerated".
The organisation has sought to slightly modify the stadium design after plans to host some Euro 2028 matches were dropped.
In November, the GAA said a reported estimate of around £270m was "closer" to the assumptions it was working on.
The Department for Communities (DfC) and the GAA have been approached for comment.
