The Daily Express says he could be named as Swansea's new manager before the end of the season.
With the Swans guaranteed a fourth straight season of top-flight football, Monk is set to meet with chairman Huw Jenkins this week to discuss taking over full-time.
Monk has plenty of support within the dressing room and boardroom, and has already held talks over potential transfer targets and pre-season tour destinations with Jenkins.
The former defender's appointment could be confirmed before Swansea face Sunderland in their final Premier League fixture of the campaign.
"We all agree that we would like to get everything sorted and clarified as soon as we can," said Jenkins.