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A new manager is required after Mark Warburton exited this month.
And they suffered a 2-1 defeat by Dundee on Sunday in Graeme Murty's second game in caretaker charge.
Further meetings with hopeful contenders are planned in London over the next few days as they look to reshape their management structure.
Ross Wilson, Southampton's director of recruitment and scouting, is one of those who has already met with Rangers managing director Stewart Robertson with a view to taking on the new director of football post.
However, there are reportedly at least another three candidates vying for the role.
Former Rangers manager Alex McLeish has said he would find it hard to turn down a return to Ibrox should he be offered another job there.
Frank De Boer, the former Rangers defender, has admitted that he would be interested but not until the end of the season.
Aberdeen manager Derek McInnes, another former Rangers player, could come into the Ibrox board's thinking with his current team six points ahead of the Glasgow outfit in second place in the Scottish Premiership.
Former Crystal Palace boss Alan Pardew, St Johnstone manager Tommy Wright and Billy Davies, the former Rangers player who last managed Nottingham Forest, are behind McLeish in the bookmakers' odds.
Rangers are 30 points behind defending champions Celtic as the Ibrox side's city rivals lead the Premiership.
They have only won one of their last four games - a Scottish Cup win over Greenock Morton in Murty's first game in charge - and they are only five points ahead of Hearts as they chase a Europa League place.
However, PA reports that no appointment for either Ibrox position is expected this week, meaning Murty is again likely to be in charge when the Light Blues travel north to take on Inverness Caledonian Thistle on Friday night.

The car was stopped in Duncrue Street in Belfast on Wednesday evening at 20.15 GMT.
Police seized suspected herbal cannabis and cocaine thought to be worth a substantial amount.
The men, aged 28 and 31, have both been arrested for possession and supply-of-drug offences.
Det Insp Tom McClure from the PSNI's Reactive and Organised Crime Branch said: "I believe that this intervention has struck a blow to the activities of an organised gang involved in serious drug criminality."

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The 49-year-old man sustained a head injury following the incident at about 09:00 on the Loan Footpath, at its junction with Fergus Avenue.
He was taken to the Western General where his condition is now considered to be serious.
Police Scotland are trying to trace the cyclist, who did not swap details with the jogger at the time.
Sgt Neil Inglis said: "We're eager to trace the cyclist, described as a man in his early 20s, and any witnesses, as soon as possible in the hope that they can help us establish the full circumstances surrounding this incident."
Police are also appealing for witnesses to a collision between a motorbike and car in East Calder.
A Kawasaki motorbike and a Kia Ceed collided at about 17:15 on Friday on the A71 junction with the B7031 Camps.
The A71 was closed for about an hour while the road was cleared.
A 41-year-old man was taken to the Edinburgh Royal Infirmary with serious injuries, which are not thought to be life-threatening.

He won the Lightweight and 350cc Junior Classic races, either side of Swallows' 16th win in the 850cc Classic.
The event, featuring bikes made before 1972, is a TT curtain-raiser and takes place at Billown Circuit.
Meanwhile, Vince Biggs and Vicky Cooke won both of the Billown event's sidecar races.
Jamie Coward, Iain Robertson and Steve Ferguson also won races at the Pre-TT Classic which has been staged at Billown each year since 1988.
Rangers have held interviews as they seek a new team boss - and to fill the new role of director of football - PA Sport has reported.
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Police say the arrests of two men following the seizure of drugs from a car in Belfast have dealt a blow to organised crime.
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Each day we feature a photograph sent in from across England - the gallery will grow during the week.
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A jogger has been seriously injured in a collision with a bicycle in Livingston.
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English rider Mike Hose claimed two wins at the Pre-TT Classic in the Isle of Man on Monday to move level with Bill Swallow's tally of 16 victories.