It's entirely possible that he has been, particularly with this individual standing up and stating that they had.
The fact that this particular person is eligible to nominate him begs the question - is that meaningful? If not, this could account for it only showing up in tabloid publications, currently, even if true.
It looks like any elected representative of any government, as well as appointed members of any government, as well as any university professor or former professor across a wide range of specific disciplines, as well as members of types of NGOs can nominate. The pool of people who are eligible to submit a nomination probably runs in the hundreds of thousands (there are over 46K elected representatives in the world, for starters)
Qualified nominators Revised September 2016
According to the statutes of the Nobel Foundation, a nomination is
considered valid if it is submitted by a person who falls within one
of the following categories:
- Members of national assemblies and national governments (cabinet
members/ministers) of sovereign states as well as current heads of
- Members of The International Court of Justice in The Hague and
The Permanent Court of Arbitration in The Hague
- Members of l’Institut de Droit International
- Members of the international board of the Women’s International League for Peace and Freedom
- University professors, professors emeriti and associate professors of history,
social sciences, law, philosophy, theology, and religion; university
rectors and university directors (or their equivalents); directors of
peace research institutes and foreign policy institutes
- Persons who have been awarded the Nobel Peace Prize
- Members of the main board of directors or its equivalent of organizations that have been awarded the Nobel Peace Prize
- Current and former members of the Norwegian Nobel Committee (proposals by current members of the Committee to be submitted no later than at the first meeting of the Committee after 1 February)
- Former advisers to the Norwegian Nobel Committee
Unless otherwise stated the term members shall be understood as current (sitting) members.
The Nobel Prize: Nomination and selection of Peace Prize Laureates
If Trump wanted to, he could order his Attorney General, or Secretary of Defense (or any other cabinet member) to nominate him, and it would be a legitimate nomination. An outspoken conservative politician doing it to troll liberals doesn't seem to be a difficult proposition to imagine.
Simply getting a nomination, itself, doesn't seem to be all that onerous of a hurdle to clear, nor especially meaningful or significant.
For context, Adolph Hitler was nominated in 1939 (to mock the Prize, itself, I think). Take from that what you will.
The Nobel Prize: Nomination Archive - Peace Prize, 1939