According to the Daily Express, the 44-year-old presenter has impressed bosses at the station on occasions when she has stood in for other DJs.
An insider commented: "There's a strong feeling more women are needed on Radio 2 and that Liza more than deserves a permanent slot.
"She could finally get her own show by next year."
Tarbuck has filled in for Steve Wright, Sarah Kennedy and Jonathan Ross on the station over the past two years.
Ross's own Radio 2 show was recently tipped to go live again following concern over the programme's falling popularity.