I agree with all the posts previously made.

The Don knew that Barzini would make his move soon and probably use someone very close to get to Michael. It was the smart move and quite likely the strategy the Don himself had used back when he was on the rise. So it's not that he was gifted with foresight but just around forty years of criminal activity had given him some insights on such things.

As to whether he should have known it was Barzini sooner, well maybe. But the Don made very few mistakes. As Tattaglia had lost a son to the Corleones, perhaps it was reasonable to ass-u-me that it was Tattaglia that would be most vigorous in prosecuting the war.

It's not until the meeting where Barzini subtly takes control of the discussion that's it's clear to Vito who his primary enemy is. After Sonny's death it could be that Vito thought things through and realized that Tattaglia didn't really have the guns or the brains to go head up against Santino, but Barzini did. Again as others mentioned he was recuperating from multiple gun shots and was preoccupied with healing...

"When the snows fall and the white winds blow, the lone wolf dies but the pack survives."
Winter is Coming

Now this is the Law of the Jungle—as old and as true as the sky; And the wolf that shall keep it may prosper, but the wolf that shall break it must die.
As the creeper that girdles the tree-trunk, the Law runneth forward and back; For the strength of the Pack is the Wolf, and the strength of the Wolf is the Pack.