Liverpool pair Henderson and Lallana are alleged to have spent a day off given to the England squad last Saturday in Bournemouth, visiting a strip club in the evening.
Henderson and Lallana both played in England’s 3-0 win over Scotland on Friday night and the 2-2 friendly draw with Spain on Tuesday, just days after their strip club trip.
But the FA will not punish either player and did not tell England captain Wayne Rooney to apologise over the “inappropriate” pictures of him drunk at a wedding that were published last week.
Southgate has been in charge of the national side for the past four matches on an interim basis – but it is understood he is the only candidate currently scheduled to interview for the permanent post.
Should everything progress smoothly, a formal appointment could follow within a week to 10 days.
Assessing Southgate’s pitch will be a FA panel comprising chairman Greg Clarke, chief executive Martin Glenn and technical director Dan Ashworth, with League Managers’ Association chairman Howard Wilkinson and former England left-back Graeme Le Saux also present in an advisory capacity.
The 46-year-old has already made a favourable impression on the decision-makers, taking seven points from nine in World Cup qualifying before signing off last week with an entertaining draw against Spain.
Southgate’s interview, which is expected to take several hours, will also include a discussion around his proposed backroom team and other staffing issues.
Chelsea coach Steve Holland has been working as Southgate’s assistant and a full-time move on to the FA staff could be in the offing, though Blues boss Antonio Conte is keen to keep Holland.
Also waiting for news is Aidy Boothroyd, who has been looking after the England Under-21s in Southgate’s absence.
Southgate has been in charge of the Young Lions since 2013 and is a keen supporter of a unified international pathway, adopting similar patterns of play throughout the age groups.
As such, he would want to work closely with any successor at under-21 level and would likely have a strong voice in any such appointment.
One issue that may not receive as much attention is the guidelines around players’ free time on England duty.
Photographs of Rooney appearing to be drunk in the early hours of a wedding party he had been invited to join during a day off at the team hotel last week have caused embarrassment for both the player and the FA.
The activities of other players, who went out in a variety of locations on the same evening, have since been placed under the spotlight and the FA has said it will review the current protocols.
That process is unlikely to get off the ground until the new manager is officially in place, with his input vital.
The usual debrief from the games against Scotland and Spain will take place separately from Southgate’s interview, later this week.