DENVER -- Mark Teixeira's injured right knee seems to be improving quicker than initially anticipated, and the Yankees' first baseman believes that he may be able to begin a Minor League rehabilitation assignment next week.

Teixeira said that a combination of cortisone and lubricant have eased the discomfort he feels from an articular cartilage tear in the knee. He is aiming to play for Double-A Trenton beginning next Tuesday, with the intention of rejoining the Yankees for a June 24 game against the Twins.

"It feels so much better," Teixeira said. "I ran yesterday and felt really good running. … The shots really seemed to work. It's all about lubricating the joints so it doesn't catch. It's worked so far."

In addition to jogging across the outfield grass at Coors Field, Teixeira did tee and soft-toss work in the batting cage. He hopes to take batting practice and ground balls on Thursday when the Yankees visit Target Field.

Manager Joe Girardi said that Teixeira's return date seems realistic, as Teixeira has been on the disabled list since June 4.

"We talked about our goal was three weeks," Girardi said. "We weren't sure which side of that goal of three weeks it would be on. There's some things that he's dealing with and you don't know how they're going to respond to the treatment and all the procedures that they go through, but so far he's responded really well."

Teixeira said that he would probably play in two Minor League games before being activated.

"You always wonder how it responds, so you play first base and hopefully run around the bases a little bit," Teixeira said. "If the knee doesn't blow up on you, then that means that you're in a good spot. If I can play two games in a row there, then I'd like to be back on [June 24]."

In Teixeira's absence, the Yankees are planning to proceed with a platoon of Ike Davis and Rob Refsnyder at first base.

Entering play on Wednesday, Yankees first basemen were ranked last in the Majors in batting average (.175) and slugging percentage (.271). Teixeira's struggles have been the major component of that; he is batting .180 with three homers and 12 RBIs in 48 games.

"Timing comes in and out during the season, and my timing has been bad all year," Teixeira said. "Hopefully, it's going to click once I come back."

Bryan Hoch has covered the Yankees for MLB.com since 2007. Follow him on Twitter @bryanhoch, on Facebook and read his MLBlog, Bombers Beat.