The Scotland international was a £10m signing from Hull City last summer and played a key role in the Reds' run to the Champions League final.
Robertson was not expected to challenge for a first-team spot so soon after his arrival at Liverpool but he quickly established himself as a first-choice player under Jurgen Klopp.
The 24-year-old has three years still to run on the deal he signed 13 months ago but, according to The Mirror, a bumper new contract is being drawn up by the Reds hierarchy.
The report also claims that Robertson will almost double his wages to more than £60,000-a-week.
Robertson has made 31 appearances for Liverpool in all competitions, as well as winning 21 caps for Scotland.