He once refused to play in an FA Cup tie for Hereford United after hearing that the club were interested in signing another goalkeeper.
News that Randolph is on his way to Fir Park comes a day after Motherwell boss Craig Brown played down his interest.
"I think it's his agent trying to promote things. That's what happens," Brown said.
"When the agent phones and asks if you're interested and you don't say no, he thinks you're going to sign him.
I didn't deny interest in him, but that doesn't necessarily mean we are signing him
Motherwell manager Craig Brown
"I didn't deny interest in him, but that doesn't necessarily mean we are signing him."
Brown has already signed one goalkeeper this summer, snapping up Lee Hollis after the 24-year-old's contract ended with Airdrie United, while Michael Fraser remains on the club's books.
However, the arrival of Randolph could be a signal that Motherwell have given up on retaining John Ruddy, who was first choice while on loan from Everton last season.
Randolph is a product of Charlton's youth system but has failed to become a first-team regular at the Valley.
He has had loan spells with Accrington Stanley, Gillingham, Bury and Hereford.
While at Edgar Street in 2008, manager Graham Turner revealed that he had replaced Randolph with Craig Samson, the former Dundee keeper, for the tie against Dagenham and Redbridge.
He said that the player had been fined two weeks' wages for refusing to take part because of Hereford's interest in Matt Murray on loan from Wolves.
The Republic of Ireland B and Under-21 international did not play for the League One club again but made 14 appearances for Charlton in the same division last season.
Source: BBC Sport
Source: BBC Sport