Reina suffered a minor hamstring injury while on international duty with Spain last week and was not risked against Reading.
Jones impressed in the 1-0 win at Anfield, with two top-class saves denying the Royals an equaliser, but Reina hopes to replace the Australian in goal in Sunday's Merseyside derby against Everton.
Reina told Liverpool's official website: "I'm getting better, little by little.
"It wasn't worth taking any risks because you risk it getting worse. I'll be patient and get really good.
"Brad was excellent. He was composed, he was really good in the air, when he was called on he was fast off his line, and he played well with his feet. He had a complete game.
"He's a good fighter and a good 'keeper for Liverpool, not just an understudy. He's another 'keeper for the club and he'll do his best to play as many games as possible."