Coveney vows to serve in role where he is 'most needed' by new leader
Mr Coveney, speaking in Cork, also said he would serve in whatever cabinet portfolio Mr Varadkar deemed to be of maximum benefit.
"I am very happy in my current brief," he said.
"We are doing a lot from a housing and strategic planning perspective.
"But I will serve where I am most needed - it is as simple as that."
When asked about whether he would like the role of Tánaiste given his strong showing in the Fine Gael leadership election, Mr Coveney demurred.
The Cork TD won the leadership vote among ordinary Fine Gael members - but lost both the council members and parliamentary party votes to Mr Varadkar.
Mr Coveney also refused to be drawn on speculation linking him with a move to the Department of Foreign Affairs.
"I make no comment on that at the moment," he said.
"I am happy where I am (Housing). Leo Varadkar needs to decide where I can be most useful in the context of his new Government."
Mr Coveney laughed when asked if Mr Varadkar might act on one of his FG election debate comments that given the Cork TD enjoyed marine matters so much he might like to return to the Department of the Marine. He had joked at the time about Mr Varadkar going back to the Department of Health.
"I am very interested in the marine - but I am interested in lots of things," he said. "I am a bit of a policy guru. But, look, we have put a lot in place in housing in the last 12 months.
"I am happy to stay to finish the job in many of the areas we are working on."
Mr Coveney also said his staff were busy preparing for the unveiling of the Ireland 2040 development plan, one of the most ambitious in the history of the State.
"But if Leo thinks he needs me in another brief, well, we will have to discuss that.
"The conversations I will have with Leo will be confidential until he is ready to make announcements."
Mr Varadkar is expected to be formally confirmed as Taoiseach next Wednesday.
He is then expected to give indications of his cabinet appointments within 48 hours.
However, Mr Varadkar has insisted he has given no indications as to his planned cabinet appointments.
Last week, Mr Coveney said he "hopes and expects" that his main cabinet backer, Health Minister Simon Harris, will either keep his position or get another cabinet portfolio.
When asked about Mr Harris, Mr Varadkar said his decision on all cabinet appointments would be based on "what is best for the country".
But the Dublin TD also said that, while taking issues of gender balance and regional representation into account, he would also give weight to "the core issue of trust".
This has been interpreted by some as a warning by Mr Varadkar that he now expects loyalty from all cabinet colleagues, including those who backed his rival in the leadership race.
"Anyone who heads up a government, heads up a football team, needs to know that everyone on that team is going to be fully behind your programme," he said.
Mr Varadkar's leadership bid was decisively supported by key cabinet figures including ministers Frances Fitzgerald, Charlie Flanagan, Richard Bruton and Paschal Donohoe.