Many believe that Prince Charles will take on this role, but, after the Queen named Prince Harry as Commonwealth Youth Ambassador on Monday, some believe that the role could go to her grandson, Prince William.
The Queen is the current head of the Commonwealth.
Commonwealth leaders are gathering in London for an executive session on Thursday and will hold a retreat at Windsor Castle, west of the city, on Friday for talks behind closed doors.
May's spokesman said Charles had proven a "proud supporter of the Commonwealth for four decades and has spoken passionately about the organisation's unique diversity". "I just don't think it should be him".
Jeremy Corbyn, the leader of Britain's Labour main opposition, suggested on BBC television on Sunday that the figurehead role could go to a rotating president.
UK Foreign Secretary Boris Johnson said the matter was a decision for "the 53". The rep declined to comment whether May believed Charles should inherit the position from the Queen. Queen Elizabeth took over the position from her father, King George VI, in 1953 during her coronation.
The Commonwealth Secretariat did not confirm if the succession would be decided this week. At the last Commonwealth Heads of Government Meeting (CHOGM) in Malta in 2015, she had endorsed for Charles to continue her role. "That enables them to deal with some quite tricky, sensitive issues, but collectively, collegiately and as part of the family", she said.
"Fifty-three heads of the governments are the heads, and they will make a decision whatever way they will determine", she told "Peston on Sunday" presenter Robert Peston.
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