The Labour Party, meanwhile, won't make a move against him for fear of bringing down the Coalition Government.
Mr Kenny is suspected of pulling an old Charlie Haughey trick of threatening a reshuffle to keep ministers on their toes – with doubts in Cabinet over whether there will be any changes before the local and European elections in 2014.
And the old guard of veteran ministers, like Michael Noonan, has no intention of clearing the way for young blood.
The Coalition's ideological row over income tax is set to continue right through to the next general election.
A Cabinet reshuffle is expected by many ministers in the autumn following Ireland's EU presidency. Mr Kenny gave his clearest indication over the Christmas period that he planned a shake-up this year.Fuelling the speculation, he said Ireland's EU presidency had provided a "reprieve" for ministers. But other ministers believe there won't be a reshuffle for the next 18 months.
"He's keeping people on their toes. It's Haughey tactics," a minister said.
"I don't think there will be a reshuffle until 2014 after the local elections, say July 2014. Then you freshen the team."
From the pool of junior ministers, only Brian Hayes and Lucinda Creighton are viewed as making a sufficient impact to be considered for promotion, although Fergus O'Dowd will come centre stage this year with the rollout of water metering.
But the prospects of veteran ministers stepping aside easily are slim. "Noonan is in his element. He looks at this as a chance to leave a good legacy."
Whenever the reshuffle does occur, the embattled Dr Reilly is set to stay put.
"The Taoiseach is very supportive of him. The understanding is that (Dr Reilly's personal finances) have substantially improved. They (Labour) won't be interfering with him. Everyone knows health is bad and a difficult department," a minister said.
Labour ministers have lost faith in Dr Reilly, regarding him as the "ultimate bull in a china shop with no ability to deliver". But the coalition partners won't put the Government at risk by calling for him to be dropped.
"Fine Gael have a lot invested in him and will do everything they can to ensure he doesn't fall on his face," a senior cabinet source said.
Down the line, Mr Coveney is viewed as the heir apparent – even though there is no sign of a vacancy arising for some time.
Following a capable under-the-radar stint as Agriculture Minister, he is well regarded.
"He has . . . left a positive impression of being well on top of his brief," a senior Labour source said.
The next six months will be vital as he has the chance to help reform the Common Agricultural Policy. But colleagues believe he would fancy a move to the Department of Communications, Energy and Natural Resources. He held the portfolio for almost a decade on the Fine Gael frontbench.
Meanwhile, Mr Gilmore and Public Spending Minister Brendan Howlin are defending the so-called war cabinet – the Economic Management Council.
The Labour members, who sit on it with Mr Kenny and Mr Noonan, say it has been successful. But Social Protection Minister Joan Burton has complained about the exclusion of other ministers in the formulation of policy.