The Three Lions boss is understood to have misgivings about having a striker as his captain.
But the goal ace Kane is edging ahead of rivals Jordan Henderson, Gary Cahill, Joe Hart and Eric Dier, who wore the armband for the November friendlies against Brazil and Germany.
Kane is in desperate need of a lift after Spurs’ midweek Champions League exit to Juventus.
Southgate is believed to have spent a great deal of time anguishing over who should take the role permanently ahead of this summer’s World Cup.
Hence his decision to rotate the job, with five players having taken it on in his 14 games in charge.
Chelsea defender Cahill and West Ham goalkeeper Hart have fallen down the pecking order due to a lack of playing time at their clubs and Southgate is well aware even Liverpool fans remain unconvinced of the merits of having Henderson as their captain.
Tottenham’s Dier is not thought to be a serious option and took the armband in November because Henderson and Kane were absent and Cahill stayed on the bench.
Southgate - who names his squad for the friendlies with Holland and Italy next Thursday - also has serious worries about the goalkeeping position.
Hart has not played a league game for the Hammers since the 4-0 defeat at Everton in November and Southgate is expected to include Burnley’s Nick Pope in his squad for the first time.
Pope has been in outstanding form since replacing the injured Tom Heaton at the start of the season.
He could be one of four Burnley players to be called up, with centre backs Ben Mee and James Tarkowski and midfielder Jack Cork also in the running.
Tarkowski is also still being courted by Poland, who he can represent due to his grandad.