The Europa League finalists confirmed the signing of the Welsh international on Friday.
Lawrence opted for a move to Ibrox following the expiry of his Derby County contract ahead of a switch to Sheffield United who are managed by Heckingbottom.
He revealed that the Blades couldn’t compete with Rangers’ “pull” following their success in Europe.
"If Macca (Stuart McCall) were here answering this question, or Flecky (John Fleck), they'd be raving about Rangers and European football,” he told the Sheffield Star.
"A team who got to the Europa League final last year, things like that, it's going to be a big pull.
"Obviously they're big payers, a big pull, different football. It's one of those things.
"You can't get them all. Would I have liked him? Yeah.
"If Rangers hadn't have come in and blown everyone out of the water I would have backed us to get him."
Lawrence became the club’s third summer signing after John Souttar and Antonio Colk.
Rangers are continuing to pursue other targets with Giovanni van Bronckhorst confirming sporting director Ross Wilson is “busy" working on other targets.
“Of course we want to add more players to our squad and he is busy talking with players, talking with agents and clubs,” he said.
Bayern Munich starlet Malik Tillman is one of the latest names to be linked with a move to Ibrox, while Welsh star Rabbi Matondo is expected to join from Schalke.