The US president, who was overwhelmingly defeated in November's election, is reportedly considering travelling to his Turnberry golf resort to avoid Mr Biden being sworn into office.
But Scotland's first minister stressed it is illegal to travel in or out of the country without a valid reason and said: "Coming to play golf is not what I would consider to be an essential purpose."
The White House has repeatedly refused to say what the outgoing president will do when Mr Biden is inaugurated on 20 January, prompting speculation about whether Mr Trump will attend the ceremony.
But Prestwick Airport has been told to expect the arrival of a US military Boeing 757 aircraft previously used by Mr Trump on 19 January, according to the Sunday Post.
Asked about speculation that Mr Trump could travel to Scotland in order to avoid the inauguration, Ms Sturgeon said: "I have no idea what Donald Trump's travel plans are, you'll be glad to know.
"I hope and expect that – as everybody expects, not everybody necessarily will hope – that the travel plan immediately that he has is to exit the White House.
"But beyond that I don't know.
"We are not allowing people to come in to Scotland without an essential purpose right now and that would apply to him, just as it applies to anybody else.
"Coming to play golf is not what I would consider to be an essential purpose."
On Monday, Mr Trump declared he would "fight like hell" to hold on to the presidency and appealed to Republican politicians to reverse his election loss when they convene this week to confirm the Electoral College vote.
Mr Biden won the presidency with 306 electoral votes to Mr Trump's 232.
He received 81,283,485 votes versus the incumbent's 74,223,744, a margin of more than seven million in the popular vote.
Additional reporting by PA
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