Allen says the relationship between the two clubs was pivotal in Kent moving to Glasgow before Monday's deadline.
Speaking to talkSPORT host Jim White, Allen said: "We made our intention very clear at the start of the window; Ryan was a key target and we got there in the end.
"Liverpool Football Club have been fantastic first and foremost in what they've done and helped us trying to get this over the line.
"A lot of credit needs to go to Liverpool for that.
"Ryan enjoyed his time at Rangers and he was obviously keen to come back, but the respect enjoyed between the two clubs helped us get him.