Burnley have made a combined offer of £25m for West Brom's Craig Dawson and Jay Rodriguez, Sky Sports News understands.
The Premier League club bid for the pair on Tuesday but it is expected their offer will be rejected.
Burnley could go back with an improved offer but any second bid would not be too much higher than their original valuation.
Sean Dyche will be looking to strengthen his team due to Burnley playing in the Europa League this season.
West Brom have already seen Jonny Evans leave the club after completing a £3.5m move to Leicester earlier this month.