It is understood that Ibrox chiefs have already started interviewing candidates for both roles.
And they are planning further meetings with hopeful contenders in London over the next few days as they look to reshape their management structure following Mark Warburton’s shock departure.
Under-20s boss Graeme Murty was again in charge at Dens Park on Sunday, but he could do little to halt Gers’ faltering run that has now seen them win just one of their last six Ladbrokes Premiership fixtures.
Paul Hartley’s Dark Blues were stronger all over the pitch than their visitors from Glasgow.
And they claimed a deserved 2-1 win - their first over the Light Blues on Tayside since 1992 - thanks to first-half strikes from Mark O’Hara and Kevin Holt.
Joe Garner did pull one back after the break but the result now means the Ibrox side trail Aberdeen by six points in the race for second.
With fourth-placed Hearts just five points behind, Gers’ plans for a return to European competition could be put in considerable jeopardy if they fail to turn around their worrying form.
And that is likely to spur Dave King’s board into action soon rather than later.
Alex McLeish remains the bookmakers’ favourite for the top job, although former Crystal Palace boss Alan Pardew has moved up the reckoning ahead of the likes of Derek McInnes, Tommy Wright and Frank De Boer.
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, but there are at least another three candidates vying for the role.
However, no appointment for either position is expected this week, meaning Murty is again likely to be in charge when the Light Blues travel north to take on Inverness on Friday night.