Midfielder Jennings joined Rovers in October and has gone on to make five appearances with Moore hopeful his stay can be extended until May.
Winger Peterson, meanwhile, is due back at parent club Liverpool after completing a one-month loan deal yet Tranmere want that arrangement to continue.
"We'd like to see if we can get (Jennings) tied up until the end of the season and then take things from there," Moore told the club's official website.
"I'll be speaking to his agent this week.
"We're still speaking to Liverpool regarding Peterson, we'd like to see that one go ahead. I need to speak to Brendan (Rodgers, Liverpool's manager) about the possibility of maybe taking him until the end of the season."
One player who could be leaving is midfielder Tom Hateley.
Moore revealed a club in Poland had shown an interest in the 24-year-old and he is unlikely to stand in the player's way.
The Tranmere boss added: "There's a club in Poland who are interested in Tom so we'll wait and see on that one.
"But it looks like he'll be off because he wants regular football and we can't guarantee him regular football here."
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