Veteran right-back Gary Neville also faces a race to be fit for the opening fixture at home to Reading.
Rossi will take a medical and be presented at the Madrigal on Wednesday. Media reports have put the transfer fee at around 10 million euros (£6.7 million), with Rossi penning a six-year deal.
The 20-year-old American-born Italian spent the second half of last season on loan with Serie A side Parma, and scored nine goals to help them avoid relegation.
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Rossi, an Italian under-21 international, joined United from Parma in July 2004 and last year signed a contract extension until 2010.
Villarreal have been on the hunt for a striker after selling Uruguayan top scorer Diego Forlan to Atletico Madrid for 21 million euros last month.
They finished fifth in the Primera Liga to earn a place in next season's UEFA Cup.
Meanwhile, United manager Sir Alex Ferguson told the club's website that Saha, Solskjaer and Neville were all "roughly a couple of weeks" away from fitness after their respective summer operations.
Paul Scholes is already ruled out of the opening stages of the season through injury.
Ferguson added that Portuguese signing Nani may miss out on Wednesday evening's friendly with Italian club Internazionale because of an ankle injury, while other arrivals Anderson and Owen Hargreaves could make their debuts.
"Nani got a little injury in his ankle in training on Monday," said the Scot. "He's a bit of a doubt for Wednesday's game."
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