The period for signing veteran free agents begins this week in the NFL, and the 49ers are expected to be active.

One position where the 49ers could strengthen themselves is at safety. It’s a deep pool of talent, with the likes of Earl Thomas, Landon Collins, Ha Ha Clinton-Dix and Tyrann Mathieu searching for new jobs.

Of all those talented players, there’s one who seems like the best fit: Thomas.

The former Seahawks star knows the NFC West, is familiar with the 49ers’ scheme (which is modeled after Seattle’s) and has been recruited by 49ers cornerback Richard Sherman, his former Seahawks teammate.

It had been presumed that Thomas’s first choice would be to sign with the Dallas Cowboys (he is from Texas and played for the University of Texas), but recent reports from Dallas indicate the Cowboys may not have the money to make Thomas the kind of offer he desires.

Recently, Sherman told Nick Wagoner of ESPN.com that Thomas would love to be a Cowboy, but he also wants to cash in on his first chance at free agency.

“Now if Dallas is a player in it, I’m not going to lie, he’s going to go (to) Dallas,” Sherman said. “That’s his home. If the money is equal, if all things are equal, he’s going to Dallas.”

But the Cowboys have a number of players on their roster they want to keep, so much of their salary-cap space could be funneled in that direction and not toward Thomas.

Plus, according to Wagoner, Thomas is hoping to become the NFL’s highest-paid safety, at about $13 million per season. The 49ers – with close to $70 million available under the salary cap – could be able to meet that price.

Wrote Wagoner: “Surely Thomas wouldn’t mind playing the Seahawks twice a year, and he has plenty of experience playing in the Niners’ defensive scheme.”

If the 49ers can secure Thomas to play free safety this week once the bidding frenzy begins, it will be one huge step forward for aiding a defensive unit that needs help at multiple positions.