With her extensive current affairs background, it is being argued that Miriam has the perfect combination of profile and credibility for the role.
And various political parties have revealed their support for the presenter.
Miriam confirmed to the Herald this morning that she "has not been approached by any party to stand" and that the story about her link to the presidency has taken "a wholly unexpected turn".
One senior Fianna Fail figure said: "O'Callaghan would be a very strong candidate," and added that her appointment would be "an incredible modern image for Ireland."
However, it might not be all smooth running for the star if she were to announce her intentions to run for the Aras.
A Labour party source said: "If she was interested in running she would have to have an argument beyond, 'I'm known'. She would have to have a cause or an agenda in mind."
Miriam said that she has received several queries about her being linked to the presidency, but said she already has her hands full.
"At the moment, I am just very busy focusing on my job, my charity work and my family," she said.