Jeremy Corbyn has said that Charles should not automatically take over, suggesting that the holder of the post could be decided "on a rotational basis".
"It will be up to the Commonwealth heads of government to decide what they want to do about this symbolic role".
Asked whether Mrs May believed Charles should be the Commonwealth's next head, the spokesman said: "This is obviously a decision that is taken later in the week, a decision taken by all the members together".
The Prince of Wales is set to be chosen to succeed the Queen as the next head of the Commonwealth by its 53 national leaders this week after he received the support of Theresa May.
While the Queen has been head of the Commonwealth since coming to the throne, the position is not automatically held by the British monarch.
While Commonwealth Secretary-General Baroness Scotland opted for a more diplomatic explanation when quizzed on the matter, saying the 53 heads of the Commonwealth "will make a decision in whatever way they determine".
The Corbyn ally said: "I just don't think it should be him".
Although Prince Charles will become King upon the death of his mother, the title of Head of the Commonwealth is not hereditary, and there have been increasing calls for it to be passed to someone else.
In the wake of the debate and to commence the Commonwealth Heads of Government week, Prince William has delivered a speech on the power of the Commonwealth and the Royal Family in London on Monday evening.
Commenting on Meghan's future role in the institution, he said: "I am also incredibly grateful that the woman I am about to marry, Meghan, will be joining me in this work, of which she too is hugely excited to take part in".