Sawyers was at The Hawthorns on Friday to watch an encouraging goalless draw with Bournemouth in Bilic's final friendly of pre-season.
He was announced as Albion's fifth signing of the summer on Saturday morning, just hours after Bilic said he was excited by the arrival.
“It’s close," he confirmed. "It’s the paperwork, I think, and until that’s done you never know.
“It looks like [it’s almost done] though. We targeted him as a player who will raise the quality in our midfield.
“I’ve seen a lot of him. People are talking about the Championship - almost in every team in the Premier League you have the players who are great on the ball - he’s one of them.
“He brings everybody into the game. I’m really happy the way our midfield played tonight, but he’s just going to add the quality, and another number of course.
“I wanted him, he is our boy. He’s coming back home and there’s nothing that can beat that.”
Albion zipped the ball around against Bournemouth, and their build-up play impressed the 5,000 fans inside the ground.
'We played good'
Bilic was pleased, but urged his side to take more chances in front of goal.
"We played good," he said. "The only negativity is that we didn't score, we had to score a few goals from these chances and combinations.
"I'm happy for a few reasons, a football game is 90 minutes but very rarely was the group down. They were compact and focused.
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"Bournemouth had to do something special to break us down - it was their quality, not our mistake - but even then we recovered with individual quality, pace, or a read of the game. That was good.
"We concentrated on the set pieces also, which is very good also, which is important in any league, especially ours.
"We were comfortable on the ball, everybody wanted the ball, we played a few combinations that we'd been practicing, we were on our front foot for most of the game, against a Premier League team.
"Some very good combinations, but the only thing I'm not happy is that we didn't score a goal, and you have to score goals.
"It's a bit of luck, it's a bit of not being confident enough, to say I'm going to score that goal, a bit of quality tonight in that last third of the pitch.
"That's a bit of a problem because without goals you have no chance."
Bilic felt Albion were sometimes guilty of overplaying in the final third, although he praised one particular scoop over the defence from Hal Robson-Kanu that set up a chance for Kieran Gibbs.
"If you're in the box I want you to shoot!" he said." Tonight they weren't that confident, they were trying to find other solutions in the box.
"I'm not talking about Hal's one, that was great - but sometimes you try and put yourself in a better position and make a pass.
"You are on the penalty spot, try (and shoot). That's the only part of the game (that was negative)."