In September 2020, two Scandinavian politicians announced separately they were nominating Trump for the 2021 Nobel Peace Prize. One of them, Christian Tybring-Gjedde of Norway, also claimed to have nominated Trump in 2018.
Contrary to a rumor circulating at the time, no Nobel Peace Prize nomination for Trump was announced in 2016.
Given that the Nobel Foundation does not reveal the names of nominees until 50 years after the fact, we can't say for certain whether Trump has or has not been nominated.
On Sept. 9, 2020, Christian Tybring-Gjedde, a member of Norwegian parliament and the conservative Progress Party, announced he had nominated U.S. President Donald Trump for the Nobel Peace Prize, for his role in cementing the Abraham Accords, which normalized relations between Israel and the United Arab Emirates. White House Press Secretary Kayleigh McEnany called the announcement “a hard-earned and well-deserved honor.” (As it happens, this wasn't the first time Tybring-Gjedde was reported to have nominated Trump for the prize. According to the BBC, he was one of two members of Norwegian parliament to put the U.S. president's name forward in 2018.)
On Sept. 11, 2020, Magnus Jacobsson, a Christian Democrat member of Sweden’s national legislative assembly, penned a letter proposing that the governments of the United States, Serbia, and Kosovo be awarded the Nobel Peace Prize, for their joint work for peace and economic development leading to a cooperation agreement signed in the White House. In effect, it was the second announced nomination of Trump for the 2021 Nobel Peace Prize.
Trump was rumored to have received a nomination in February 2016 (before he was elected president), when Facebook users were surprised by an unusual trending topic holding that he was among the Nobel Peace Prize nominees for that year:
The claim's appearance on Facebook's trending topic list and fairly definitive wording led many to believe the assertion was without ambiguity: Somehow, Trump was in the running for a Nobel Peace Prize. However, such accounts were based on a very loose definition of "nominated," one that relied upon a "Nobel watcher's" report of a single letter which put forth Trump's name (and was quite possibly satirical in nature):
The five-member Norwegian Nobel Committee typically receives more than 200 nominations for the prestigious prize and keeps candidates secret for 50 years. Panel members can make their own nominations during their first judging meeting on Feb. 29.
Nobel watcher Kristian Berg Harpviken, the director of the Peace Research Institute of Oslo, told the French news agency AFP that he had received a copy of the nomination letter, suggesting Trump — who drew criticism from around the world for his vow to ban Muslims entering the United States — should be awarded the prestigious prize for "his vigorous peace through strength ideology, used as a threat weapon of deterrence against radical Islam, ISIS, nuclear Iran and Communist China".
As such accounts noted, Harpviken maintained he'd seen a single letter nominating Trump and was of the opinion that the potential nominee had no real chance of winning the Nobel Peace Prize. Nonetheless, many social media users indicated they believed that Trump had been formally nominated for the 2016 prize and/or was seriously in the running to be awarded it.
In general, anyone's name can be suggested for a Nobel Peace Prize, and more than 200 such submissions are made annually. Given that the Nobel Foundation does not reveal the names of nominees until 50 years later, we cannot say conclusively whether Trump has been one of them.