The 28-year-old is expected to take over from Kepa Arrizabalaga, who the Blues made the most expensive keeper in the world when they signed him from Athletic Bilbao for £71.6m in August 2018.
Kepa has made two errors leading to goals in as many Premier League appearances this season, though, but will not be displaced just yet.
"To start tomorrow is not going to happen," Lampard said when asked if Mendy was in contention to start at the Hawthorns on Saturday.
"He's had a long week where he hasn't really trained, flew in yesterday and it just feels like too tight of a turnaround to put him in contention to start the game."
Asked if he had spoken to Kepa since Mendy's arrival was confirmed, Lampard replied: "I haven't spoken to Kepa since then no. The announcement came and we trained and I didn't see Kepa, but I will have a conversation with him and Willy Caballero over the next 24 hours or so.
"I have a very open line of communication with both of them. It's important as goalkeepers, they have their own small union in the training ground because of the nature of the way they work together.
"It needs to be a very positive feeling between the goalkeepers and I'm sure there will be. There's good competition there now, we know that and it's how we move forward."
Chelsea also secured deals for Timo Werner, Hakim Ziyech, Kai Havertz, Thiago Silva, Malang Sarr and Ben Chilwell prior to the 2020-21 season.
Antonio Rudiger and Jorginho have consequently been linked with exits from Stamford Bridge, but Lampard was giving nothing away on their futures.
"I don't expect anything in the window, because they're our players and that's a fact. There are lots of rumours about them and others," said the Chelsea boss.
"Every player will always be a case-by-case solution, that would be a happy solution for the player, the club and myself. Yes, we have a big squad, we know that, but at the moment players remain ours and we'll see as it goes on."
Lampard was similarly guarded when questioned on whether Declan Rice could be bought from West Ham before the transfer deadline on October 5.
"Again, they're rumors. I can't talk about these things but I know it's your job to ask," he said.
"When players are ours it's not my position to speculate on what maybe because that's not my job, and the same on the other side with players that are playing for other clubs.
"We have a game against West Brom. My biggest thing is about getting the squad ready for that game."