The Red Devils have already failed to land the player with two bids, but latest reports claim Woodward is now close to meeting the £48million asking price.
United had been keen to bring the Croatia global to Old Trafford this summer but, while Sky sources understand they were close to agreeing a fee with the Serie A side, no deal was made.
Although a deal has stalled as both clubs fail to agree on a fee, United have not ended their interest in Perisic and will look to sign him before the end of the window.
Speaking after Inter beat Real Betis 1-0 in a friendly on Saturday, Spalletti told Sky Sports Italia: "Perisic is determined to remain".
The Manchester Evening News reported last week that United were ready to renew their interest in the Croatian worldwide after seeing two previous bids rejected by Inter.
Perisic has registered a not-too-shabby 26 goals and 22 assists in 112 games for I Nerazzurri, and his versatility in being able to play anywhere across the forward line would be a huge bonus to Barcelona's tactics of playing with a fluid and dynamic attack.
"They're strong, they put up a good defence to keep someone who's an important player for them", he was quoted as saying via Daily Express.