A suspected gunman arrested near Donald Trump's golf-course wrote a note months earlier saying he intended to kill the former president, a court filing shows.

"This was an assassination attempt on Donald Trump," the note says.

Prosecutors included the note in a court filing on Monday, and said they would try to charge 58-year-old Ryan Routh with the attempted assassination of a major political candidate.

Routh has been in custody since his arrest on 15 September in Florida.

US Federal Judge Ryon McCabe ordered that Routh remain behind bars without bail pending his trial, saying that the "weight of the evidence against the defendant is strong".

Routh is so far facing two federal gun crime charges, including for possessing a firearm as a convicted felon.

Federal prosecutors, however, said in court that they would ask a grand jury in the coming days to also charge Routh with the more serious crime of plotting to assassinate a political candidate, saying that evidence suggests he had been plotting an attack on Trump.

In the pre-written letter, addressed to "The World" and sent to an unnamed witness months before the 15 September incident, Routh appears to pre-empt a failed assassination attempt on the former president.

"I tried my best and gave it all the gumption I could muster," the letter reads.

Routh, who told a judge during his first court appearance last week that he had no funds and no savings, then says in the letter that he would pay a cash reward to anyone "who can complete the job".

A box containing the letter - as well as ammunition, building materials, tools and four phones - were dropped off at the home of the witness before the incident, according to the court documents filed by prosecutors.

In the filing, prosecutors state that the witness, whose relation to Routh is unclear, opened the box after learning of the apparent assassination attempt and then contacted the authorities.

The documents were filed in support of Routh's continued pre-trial detention.

Routh was arrested after a Secret Service agent spotted his face in foliage while securing the sixth hole of Trump's golf course in West Palm Beach before noticing a rifle, prosecutors say.

The agent then jumped out of his golf cart, drew his gun and fired after seeing Routh allegedly move his gun, the court filing says. Routh did not fire his gun at any point during the incident, police have said.

The suspect managed to flee and left the weapon and some other items at the scene. He was arrested shortly afterwards after a witness spotted him on the Interstate 95, a major highway.

Court documents indicate he had 11 rounds of ammunition, one of which was chambered in the rifle.