The defensive midfielder, who is the brother of Watford star Nathaniel, looks likely to leave Chelsea this summer.
The Express & Star confirmed last week Albion have held talks with Chalobah as they work to bolster their squad ahead of the 2021/22 campaign.
It is understood those talks have been positive with the Baggies still firmly in the race to land the former England under-21 international.
But it is understood they face competition from a host of other clubs.
One of those sides is in the Premier League and is believed to Brentford, who won promotion via the Championship play-offs last season.
French side FC Lorient – who Chalobah enjoyed a loan stint with last year – are also understood to want to sign him permanently.
It is believed the former Huddersfield and Ipswich loanee also has interest in Germany.
Central midfield is a key area Albion need to strengthen this summer.
The Baggies have snapped up Alex Mowatt – who shone under head coach Valerien Ismael at Barnsley last season.
They also have Jake Livermore and Romaine Sawyers.
But while they played key roles in the club’s promotion under Slaven Bilic, both struggled for playing time last season.
Chalobah would also be a shrewd addition for Albion because he can also play in central defence.
And with Ismael a boss that has tended to play with a back three – that versatility could prove hugely beneficial.
Meanwhile, former Albion winger Kyle Edwards is on trial with Reading.
The 23-year-old played 56 minutes for the Royals in their 1-0 win over Lincoln City in a pre-season friendly on Saturday.
The former England under-20 international has been on Celtic’s radar for a number of years.
But the forward, who was released by the Baggies at the end of last season, may now be set to continue to his career with one of Albion’s Championship rivals.