Plymouth are currently bottom of League One and have lost their past five matches in all competitions.
They have won just four out of 15 league games so far under former Watford boss Cleverley, who was appointed in the summer.
The ex-Manchester United midfielder said following their 3-1 defeat by Huddersfield on Saturday that he will “fight” to turn the team’s fortunes around.
The club’s former head of recruitment Tom Randle – who oversaw the summer transfer window – left the club by mutual consent last month.
Since leaving Plymouth five years ago, Adams has had three stints as manager of Morecambe during which he guided them to promotion to League One.
The Scot also spent eight months as manager of Bradford City in 2021-22 and resigned after 12 games as boss of Ross County in February 2024 following a third spell in charge at the Scottish Premiership club.
“Derek is someone who knows Argyle well after a long period of time here as manager,” Plymouth chairman Simon Hallett said.
“He has extensive knowledge across all areas of the football pyramid in both England and Scotland and will provide valuable support to both David and Tom as we look to come through this challenging period.
“He is someone that has experience of tough circumstances and will work closely with all areas of the men’s football department to ensure results and performances improve.”
