The Knicks and Grant Hill are serious about each other. Donnie Walsh and Mike D'Antoni spent nearly all day -- and night -- with the former All-Star yesterday, culminating with dinner in the city and a offer on the table.

The offer was for one year and a portion of the Knicks' mid-level exception. Hill is permitted to sign as soon as tomorrow, the first day free agents are allowed to sign contracts

We think we have a chance, but you never know," a team source said last night. In a visit that blew Jason Kidd's away, Hill arrived yesterday at the Garden at 11:30 a.m., got the arena tour and the jersey before being whisked to the Knicks' practice facility in Westchester.

Last night, Hill, his agent Lon Babby, Walsh and D'Antoni headed out for dinner. The Knicks could blow the Celtics out of the water since Boston is using their $5.8 million mid-level exception on Rasheed Wallace and can offer Hill only the $1.9M lower exception.

With Kidd agreeing to terms with Dallas, the Knicks have their $5.8 million mid-level exception to use and can take half of it to top Boston on a one-year deal for Hill, who played 82 games last season, averaging 12 points and 5 rebounds.

"You want to have players who've had a lot of success before so they can show the other players how to do it," Walsh told The Post. "That's the kind of player we want to bring in."

On the surface, small forward is not a positional need, but Walsh feels Hill, at 6-foot-8, also can play shooting guard -- a void since Cuttino Mobley's retirement.

Hill also is considering staying in Phoenix, but the Suns won't outbid the Knicks. Hill told reporters yesterday at the Garden playing "at the mecca" has always been appealing to him and he has always loved New York. He claimed he would've heavily considered the Knicks when he was a free agent in 2000 had the Knicks not tried to move his buddy Patrick Ewing in a sign-and-trade. Hill would have to change his number as he wears 33, which the Knicks have retired.

"Hill's a class act," D'Antoni said. ""He knows how to play. I think he's one of the best free agents out there."