The 18-year-old centre-back played for Rovers in the Championship in a 1-1 draw against Stoke City on Saturday.
He is with them for the second half of the campaign, while Elliott has been with Rovers for this entire season.
Branthwaite told Lancs Live: "I spoke to Harvey before I signed and he said that he's loving it here, so that was another thing that made me want to come.
"He's enjoying his football and obviously he's getting lots of game time so that's a benefit for him as well.
"I thought it [the move] was perfect, to be fair. I spoke to the manager numerous times and the things he was saying about the club and how they want to play, it just fit right in.
"And with me having a house in Liverpool it's perfect for me to get into training, it only takes 35 minutes so it all just fit perfectly really."