The former Reds captain is currently 2/1 with SkyBet and Bet Victor to succeed Sam Ricketts, whose departure for Shrewsbury Town was confirmed by the club on Monday.
Lawrence, who made 198 appearances for the Reds in a five year spell between 2001-2006, has been slashed from odds of 8-1 to take the role.
Currently in charge of his native Trinidad and Tobago, the 44-year-old has moved ahead of current assistant manager Graham Barrow whose odds are currently 3/1.
The two coaches have worked together previously at both Wigan and Everton under Roberto Martinez.
Club officials have assured fans that they will move swiftly to fill the managerial vacancy, with a slew of other potential candidates available at longers odds.
A statement issued following Ricketts' exit said: "Wrexham AFC are in a strong position after 22 league games of the 2018-19 season. We would like to thank Sam for his efforts and for the work he did while at the football club.
“We are fully focused on further improving upon this situation going forward.
"Wrexham AFC have already begun the process to appoint Sam’s successor with Graham Barrow taking charge of the first team on an interim basis.
“We will make further announcements regarding this matter as soon as possible.”
Recently-sacked Shrewsbury boss John Askey is available at 6/1, with the likes of Kevin Nolan and former Reds striker Andy Morrell priced at 8/1.
Darren Ferguson is another ex-Red in the running, with former Ebbsfleet boss Daryl McMahon available at the same price of 12/1.
A wealth of experienced managers comprise the mid-market options, with Russell Slade (14/1), Phil Brown (16/1) and Gary Bowyer (16/1) all in the running.
Outsiders for the role include the more youthful Craig Bellamy and Harry Kewell (both 20/1), while Ian Holloway, Owen Coyle and Russell Wilcox available at a healthy 25/1 price.
Wrexham will hold a press conference on Friday morning to provide further updates ahead of their home clash with Eastleigh.