Rossoneri chief Adriano Galliani has revealed that the club were unwilling to meet PSV's €20m (£17.2m) valuation and has also ruled out signing a new striker this summer
The Dutch midfielder, 23, sealed a five-year deal to join the Giallorossi from PSV on Wednesday.
And Galliani admits that Milan pursued Strootman but claims his side were not willing to offer up the eight-figure sum demanded for the 23-year-old.
"Strootman is a very strong player who we had tracked for a while, but we did not have €20m for a player like him," the Rossoneri vice-president told Sky Sport.
"Roma did a great deal, though, as he is a player of the very highest level."
With the likes of Mario Balotelli, Giampaolo Pazzini, Stephan El Shaarawy and Robinho available in attack, Galliani also ruled out any summer moves for additional forwards.
“We are not looking for any strikers, because we already have players like Giampaolo Pazzini in the squad," he added.
Milan finished third in Serie A last term, while they also reached the last 16 of the Champions League before being knocked out by Barcelona.