Mr Gilmore denied Education Minister Ruairi Quinn believed Dr Reilly was "not up to the job" – and neither was this the opinion of Labour cabinet ministers.
However, neither the Tanaiste nor Mr Quinn denied he actually made the comment at a Labour Party meeting.
And another Labour Party minister privately said Dr Reilly's performance is a constant source of division in the Coalition.
"It is mentioned at every meeting of every week. Ten times a week at least. Everyone within Labour believes it is time for him to go," the minister told the Irish Independent.
"He is bumbling from one issue to the next. Where will it end?
Mr Gilmore said he saw Mr Quinn's reported comments.
"They are working as a team of ministers within the department of health. There is a very difficult job of reforming our health service that has to be done."
"They are getting on with it. They are working together and have the full support of the Government," he added.
Mr Quinn's spokesperson said the comments were "allegedly" made at a Labour parliamentary party meeting.
"As you know, those proceedings are confidential," the spokesperson said.
Fine Gael TDs hit back at Mr Quinn, pointing to the controversy over the delay in the processing of student grants.
"There is only one area of new competency in his department and that is SUSI (the new college grants application system). That's a shambles. He is in no position to be questioning the competency of another minister," a party backbencher said.
Mr Gilmore said there was no need for him to speak to Mr Quinn about the comments attributed to him.
"I know they are not his views," he said.
The Tanaiste also said the Budget was going to be "difficult".
"It is very difficult and it is painful.
"I think we have to be absolutely straight with people about that. There are a number of things that we need to take on board. When this Budget is done, that is 85pc of it (the fiscal adjustment) done and the end is in sight.
"I think we have to look at the Budget as something that is necessary.
"It is about making sure that our state finances are stable and are well managed so that we can get on with getting the investments and jobs."