The incident, the subject of police enquiries, occurred at full-time after Celtic had scored a late equaliser in the 1-1 draw at Broadwood Stadium.

McPherson, who describes his behaviour as "out of character", says he is "very grateful" to Alonso for accepting his apology after a "lengthy conversation earlier in the week".

The Rangers coach is also facing a charge from the Scottish FA and says he will "accept in full whatever punishment they deem appropriate".

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"I would like to offer a full and unreserved apology for my actions at the end of Monday's Old Firm match at Broadwood," a statement from McPherson read.

"I know I have severely let myself, our team and the club down. I hold myself to the highest of standards and have never done anything like this before in a long coaching career, and I know I fell way short of that with my actions.

"This club is also built on incredibly high standards, and I have already apologised to the board, the football staff, and our fantastic group of players who I care about dearly.

"I would also like to apologise to the Rangers fans as I am aware of what a privilege it is to represent this club in any capacity, and I am sorry that on this occasion I did not uphold the standards this club demands.

"I also understand how exciting a time it is for the women's game in Scotland, which I have come to care passionately about, and I apologise for any negativity my actions have brought in our attempts to grow the game."

The incident is being investigated by Police Scotland and Rangers.

Alonso: I was called a little rat

Rangers looked set to secure a vital victory at Broadwood until Caitlin Hayes' 99th-minute equaliser ensured the rivals would share the points.

Alonso, who has previously worked under Ronald Koeman and Mauricio Pochettino at Everton and Southampton respectively, told Sky Sports he was called a "little rat" as the footage was shown to him after the match.

The 46-year-old Spaniard said: "I don't know. You can see there, somebody pushed me from behind. I never talked to [McPherson] the whole game.

"It's obviously disappointing to concede a goal in the last minute, I totally get it. But I don't know.

"I was called a 'little rat', I don't know why."

When asked about the incident after the match, Rangers head coach Malky Thomson told Sky Sports: "Without me seeing it at all, I don't know that I can comment on it.

"If that's the case, then there will be an investigation and we'll obviously look at it."

Former Morton and Falkirk midfielder McPherson, 52, had spells as assistant manager at Morton and Dumbarton before joining up with Thompson in the summer of 2021.

The 1-1 draw - in the first SWPL league game televised live by Sky Sports - ensured Celtic and Rangers remained second and third, respectively.

Glasgow City are top of the table, eight points clear of Alonso's side, with nine matches to play this season.