In the first concrete signs of the country's preparations to leave the European Union (EU)c, May told the current council President Donald Tusk during a phone call that the UK would relinquish the role, Sky News reported.
The council is the body that sets the priorities and general direction of the EU and it is made up of the heads of state from each of the 27-member nations.
However, May told Tusk that because of the Brexit vote it would be inappropriate for the UK to take its turn.
A Downing Street spokesman said the Prime Minister told Tusk that she thought it was "the right thing to do given we will be very busy with negotiations to leave the EU", Sky News said.
The call was May's first conversation with Tusk since she became the Prime Minister earlier this month and the European Council president welcomed her quick decision on the issue.
The spokesman said that Tusk asked May for a "velvet divorce".
The Prime Minister travels to Germany later in the day for what she has described as frank and open talks with her counterpart Angela Merkel.
Speaking ahead of the trip she said she was "determined that Britain will make a success of leaving the EU" and her visits to Germany and France on Friday was to "forge strong working relationships", Sky News noted.
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