McPherson, 52, approached Alonso after a dramatic 1-1 draw in the Scottish Women’s Premier League on Monday.
He has been cited by the Scottish Football Association and says he will “accept in full” any punishment.
“I would like to offer a full and unreserved apology for my actions,” McPherson said in a statement.
“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 those with my actions.”
Celtic had rescued a point with the last kick of the match by Caitlin Hayes to remain second in the SWPL, eight points behind leaders Glasgow City.
As players and staff were shaking hands, McPherson butted Celtic boss Alonso from behind.
McPherson said he had apologised to Alonso after a “lengthy conversation” this week, and expressed regret “for any negativity my actions have brought in our attempts to grow the [women’s] game”.
He will face a Scottish FA hearing on 13 April.
Source – BBC Sport
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