Mark Teixeira hoped to be back by May 1, which has come and gone.

Now the Yankees first baseman, while reluctant to establish a time frame for a potential return, believes he will be back before the start of June.

“Absolutely,” Teixeira said, before giving himself some wiggle room. “I hope so, but we’ll see.”

His hesitation is understandable given his trouble recovering from an injury to a tendon in his right wrist suffered in early March while he was getting ready for the World Baseball Classic.

Teixeira has finally begun doing soft toss and hitting off a tee.

“It feels really good,” said Teixeira, who had his third straight day of tee-and-toss work yesterday. “We’re working up to full-speed toss, and then I’ll go outside and hit real batting practice.”

That next step is expected to happen today or tomorrow, before Teixeira heads to Tampa following tomorrow’s series finale against the A’s.

“There’s rust and maybe weakness, but no pain anymore, no stiffness,” Teixeira said. “That’s what we were working on last week. ... I feel like I got over the hump with the last few tee-and-tosses. It’s been really good to finally let it go.”

But he still has plenty more to do before he’s ready to rejoin the lineup, which is one reason general manager Brian Cashman declined Thursday to assign a date for when Teixeira might be back in The Bronx. Hitting coach Kevin Long is encouraged by what he has seen.