The former England manager is believed to head a shortlist of up to six names for the post, although sources at the Football Association of Ireland stressed that they are in no rush to make an immediate appointment.
The Irish are not due to play again until 1 March and, although no opponent has so far been scheduled, there have been preliminary discussions over a fixture with England.
The FAI wants a successor to Brian Kerr, who did not have his contract renewed after failing to qualify for next year's World Cup. However, it is thought that it favours a high-profile name such as Venables
It may be that someone like Venables, now 65, is appointed in tandem with a former player who can be groomed as his successor. Steve Staunton or Niall Quinn might fit the bill.
A salary of up to €1m (£677,000) has been proposed for the new manager. Although that would not attract either Sir Alex Ferguson or Martin O'Neill, it may be acceptable to Venables. It is thought Venables is interested in the job, but sources close to him said last night it "would be unfair to comment at this stage".
Bookmakers believe he is in the frame and suspended betting after receiving several large wagers, many from Ireland.