The cabinet meeting was Mr Howard's and Mr Costello's first face to face meeting since making contradictory claims about a leadership deal between the two men.
Mr Howard says he and his treasurer continue to disagree over whether a leadership handover deal was struck at a meeting 12 years ago.
But Mr Howard says the two men will continue to discharge their ministerial responsiblities.
The prime minister says it has been a difficult few days for the Liberal Party, and expects the political damage to be felt in the opinion polls.
The leadership issue was discussed at the cabinet meeting, and again later in a private meeting between Mr Howard and Mr Costello.
The prime minister says he feels no personal hostility towards the treasurer.
"I do not believe that the professional relationship between Mr Costello and me has broken down," Mr Howard said.
"We have worked together in the past over a long period of time. It has been a very successful partnership and I intend to continue to work in a professional manner".
A former defence minister, Ian McLachlan, has revealed he witnessed a meeting between the two men in 1994, where Mr Howard said if he won office he would hand over the leadership after two terms.
Mr Howard insists there was no deal, but Mr Costello says Mr Howard gave an "undertaking" at the meeting to pass-on the leadership midway through his second term as Prime Minister.