Reports claim Pulis, who is now in charge at Middlesbrough, is keen on being reunited with the duo on the Riverside.
The Welshman’s chances of landing either man would almost certainly depend on winning promotion to the Premier League with Boro, who currently sit sixth in the Championship with eight games remaining.
Both are players Albion, who appear destined for the drop, would be keen to keep as they look to regroup in the second tier.
Brunt recently triggered a 12-month appearance-based clause in his contract, which had been due to expire in June.
The 33-year-old is in his 11th season at The Hawthorns and recently stated his desire to reach 400 appearances for the club.
Dawson, who began his career in non-league, must meanwhile be regarded as one of the Baggies best bargain buys of the past decade, having made more than 170 appearances since joining from Rochdale in 2010.
Several Premier League clubs are thought to be keeping tabs on the 27-year-old, who can play at both right-back and centre-back and has been one of the club’s most consistent performers in recent years..
His existing contract with the Baggies runs until the summer of 2020.
Pulis took charge at Boro in December, a month following his sacking at The Hawthorns after almost three years at the helm.