Local historian John Mullaney has said that they have a very good idea of where Henry I is buried, although he might not be there anymore. He would have had quite an ornate tomb, probably with an effigy, which probably existed as late as 1400. However, now a playground sits on the site of where his tomb would have been. Perhaps Henry will become the king in the playground.
Constance of York was buried with the Henry I in Reading, though the search doesn’t seem to include her. I wrote a blogpost on this, which you can read here.