William Shatner apparently pitched a two-part episode to the creator of Star Trek: Enterprise which would have saw Captain Kirk appear in the 22nd Century.
Writer, producer and creator on Star Trek: The Next Generation, Star Trek: Voyager and Star Trek: Enterprise, Brannon Braga, tells CBR (via Blastr) that a crossover with Captain Kirk and Star Trek: Enterprise was in the works.
The only crossover that was exciting was there was a brief time when we were going to put Captain Kirk on Enterprise, and we even met with William Shatner, but it just never happened. We had some story concocted about why Kirk was there and how he got there. I don't remember. I think Shatner had a pitch. It was actually going to be a pretty cool two-part episode. I don't know exactly what happened. It might have been that we couldn't make a deal with Shatner or something like that.
William Shatner would have joined the likes of Brent Spiner as previous Star Trek actors appearing on Star Trek: Enterprise; Spiner appeared as Data's ancestor, Dr. Arik Soong.
And if you are wondering if there is any chance of seeing William Shatner as Captain Kirk or any other of the Star Trek original cast members in the upcoming J.J. Abrams Star Trek sequel, Star Trek Into Darkness, there has been more than one time where the answer has been not so.
While a new Star Trek is currently not television, you can catch "new episodes" of Star Trek: The Next Generation in the form of the Star Trek: The Next Generation: Hive comic book from IDW Publishing and Brannon Brago.