The experienced defender joined the club from Macclesfield at the beginning of February and has made 11 appearances to date.
Prior to his arrival at the Marine Travel Arena, Bunney had a brief spell at Marine's Northern Premier League rivals Ashton United.
However, his stint with the Robins was cut short to allow him to make the move to Merseyside.
Neil Young's Mariners are currently three points outside the play-off places ahead of Saturday's home game against Nantwich Town.
Photo credit: Susan Nugent
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