The prince denies the allegations.
Legal staff were waiting for Andrew to go on his regular horse ride at Windsor Castle just hours before the sex abuse allegations were filed, the Mirror reported.
They reportedly wanted to serve him court papers in person.
A source told the paper: “Virginia’s lawyers are under no illusions how difficult it is to get alongside the prince. They’ve been trying to speak to him for five years.
“Processors were on the ground on Monday when the papers were filed in Manhattan hoping to hand them to him if he went out on his horse. He could be at Balmoral for months without coming out.”
Under the Hague Convention, Virginia Roberts Giuffre’s legal team must give the papers to Andrew in person.
They have 120 days to get them to him or his lawyers could call for the case to be dismissed.
Once the papers are served, Andrew has 21 days to respond.
Her lawyer David Boles added: “We’ll present a copy of the complaint in a formal way. He is a foreign citizen so we must do this under the Hague Convention.”
It comes as a witness says he is prepared to swear on oath that he saw Prince Andrew groping Virginia Roberts Giuffre on Jeffrey Epstein’s “paedo island”.
Steve Scully, Epstein’s former telecoms specialist, says he spotted the Duke kissing and grinding against a bikini-clad blonde by a pool “between 2001 and 2004”.
He insists the woman was Virginia — and is willing to give evidence in court for the first time.
Her lawyers were understood to want to speak to Steve, 71.
He told The Sun on Sunday: “I was working on the phone and he (Andrew) was by the pool with Virginia. As soon as I saw him, I recognised him.
“I thought, ‘If I had a camera in my bag right now, I’d have a $50,000 picture’.”
Steve says they were “bumping and grinding” for several minutes before lying on lounge chairs by the pool.
In 2015 court papers, Virginia described an “orgy” on Little St James in the Caribbean with the prince.
Steve, whose story was first revealed by The Sun on Sunday last year, says he later met Andrew, now 61, as the royal held hands with the woman on a pathway.
He is convinced it was Virginia and was struck by how young she was, saying: “She stood like a kid would stand behind a parent.”
Steve added: “Andrew’s protecting himself and his family.
“If Virginia’s lawyers call, I will definitely speak to them and I would give a deposition under oath telling them what I saw.
“There’s no question in my mind that I saw Prince Andrew and Virginia together and no question that he would remember the incident either.
“He should come clean.”
Lawyers are also thought to be keen to trace mystery Epstein associate “Banu”, believed to be investment worker Refia Banu Kucukkoylu, 49.
Andrew is alleged to have forced Virginia to “engage in sex acts against her will” at Epstein’s Manhattan mansion in 2001, the lawsuit claims.
It followed an alleged incident in which he sat with her and Johanna Sjoberg on his lap and groped her with his Spitting Image puppet.
Virginia’s lawyer David Boies hinted this week that hairstylist Johanna, who lives in Florida, would provide testimony.
Flight records show that Banu, under the initials BK, was on the same plane as Epstein, Virginia and Johanna which landed in New York before Andrew’s visit.
Banu is thought to have fled her LA base for Turkey, where she has family connections.
There is no suggestion she was involved in illegal activity or knew of Epstein’s sex trafficking operation.
Voiceover artist Emmy Tayler, 46 — a former assistant of Epstein’s lover Ghislaine Maxwell — is also likely to be the focus of Virginia’s lawyers.
Emmy, of Oxford, was on Epstein and Maxwell’s March 2001 trip to London — where Virginia says Andrew sexually abused her at Maxwell’s home.
Andrew denies all Virginia’s allegations.
He has remained silent since she filed her lawsuit in New York on Monday.