John Lennon

I saw two pictures of Alice and John Lennon. They were both sitting at a table and it looked like they were signing autographs.

That sounds like the photos that were taken at a New York City night club in 1974. Alice and Lennon hung out several times back then. Especially when Lennon had his "Lost Weekend" in LA and was getting drunk with the Hollywood Vampires every night. Alice says that Lennon used to stop by the studio everyday when they recording 'Elected' to listen to the tapes of the song. Evidently, Lennon really loved the song and would tell Alice that the only thing that would have made it better is if Paul had been singing it.
(Renfield, November 1996)

There was an advert (possibly in 'Creem') advertising and early ACG album which featured the line "Alice Cooper is one of John Lennon's favourites".

In 1996, Alice was considering recording a Lennon Song ('Cold Turkey' was mentioned) for a Lennon Tribute album but it never eventually happened.

Wasn't Alice rumored to be doing a track for the Lennon tribute?

Yes. I believe I started that rumor. In the end it was just decided not to participate (I don't think that it's doing that well) and to debut Alice on Hollywood with his own album. Also, at the time, it was planned that Alice would do the tribute song with White Zombie who also wound up not participating.
(Renfield, November 1995)