Head coach Darren Moore has decided to use the international break to run the rule over the pair, who were both released by Norwich City this summer.
Playmaker Hoolahan, 36, and centre-back Martin, 32, are training with the Baggies today and yesterday.
If either of them impress they will be invited back to the Palm training ground next week for a second trial ahead of Albion's trip to Birmingham City.
Moore is looking for more options in attacking midfield after being left frustrated in his pursuit of Blackburn Rovers midfielder Bradley Dack last week.
The Baggies approached Tony Mowbray's team with an offer for Dack after selling Nacer Chadli to Monaco for £10million but Rovers told them the 24-year-old, who won League One player of the year last season, was not for sale.
Moore is now contemplating bringing experienced playmaker Hoolahan in as back-up to 20-year-old Harvey Barnes, who has impressed at No.10 in recent weeks after being moved inside off the left wing.
The Baggies boss is also keen to add defensive cover because he has just four senior centre-backs at the club vying for three positions in his back-line.
An injury to Kyle Bartley, Ahmed Hegazi, Craig Dawson or Tosin Adarabioyo could leave his defence worryingly thin, although 22-year-old Kyle Howkins did recently make his debut in the Carabao Cup and Moore has also played Sam Field in a back three.
Hoolahan spent 10 years at Norwich, and grew into a cult hero during that time, scoring 53 goals from 354 appearances.
Norwich sporting director Stuart Webber said at the time: “It was a really difficult decision. What he’s done for the club, both on and off the pitch, is immense.
"It’s always difficult especially when it’s an end of an era for somebody and I’m just glad he’s going to get the send-off he deserves, and he’s been granted a testimonial which is great."
Republic of Ireland's Hoolahan announced his retirement from playing international football in February this year.
Martin was also a long-serving Canary, and spent nine years at Carrow Road.
However, his appearances were limited last season and he spent the second half of the season on loan at Rangers.