Johnson scored his first goal for England on Friday night.
Bruce revealed: "Adam is a terrific player and we would have loved to have had him here.
"We were incredibly close in getting him on board last season.
"We spent months working on the deal, and for a while I was convinced we had him - the lad himself was convinced he was coming here.
"Then you have a club like Manchester City coming in and it is always going to be difficult to compete with that.
"In the end, they blew us out of the water, but you cannot be bitter about these things.
"I have been at Sunderland a year and Johnson was one of four or five big players we were within a whisker of getting, only for it to fall through."