The 28-year-old is a free agent after leaving Hillsborough at the end of last season and is among several targets being tracked by the Baggies. Reach, who made more than 200 Championship appearances during five seasons with the Owls, is also attracting interest from Blackburn.
Albion are likely to make multiple additions to their squad before the transfer window shuts at the end of next month but the extent of their business will be determined by the futures of Sam Johnstone and Matheus Pereira.
Both will be allowed to leave this summer, with the Baggies hopeful of receiving significant fees which can then be used by boss Valerien Ismael to fill holes elsewhere.
Interest from West Ham in goalkeeper Johnstone, however, appears over with the Premier League club have yesterday signed Paris St-Germain’s Alphonse Areola.
The Baggies rejected a bid of around £6million for Johnstone from the Hammers earlier this month, as it fell some way short of their £12m valuation of the England international.
A second bid is now not expected with Areola joining the London club on loan for the season.
Johnstone is due to return to training with the Baggies early next week and it remains to be seen whether the 28-year-old will be ready to feature in next Friday’s season opener at Bournemouth.
Tickets for the match at the Vitality Stadium went on sale to away members this morning. Albion have been given an allocation of 1,200, with tickets priced at £30 for adults and £17 concessions.