Frampton defended his IBF title with a fifth-round stoppage of Chris Avalos at Belfast???s Odyssey Arena on Saturday night with long-time rival Quigg watching at ringside.
Frampton is open to a bout against the WBA champion but insisted he deserved the upper hand in negotiations after extending his perfect record to 19-0.
???I would love it, I think Scott would too,??? Frampton told ITV1.
???A summer showdown between the two of us is a huge fight. But Scott needs to remember who the real world champion is.
???I would leave Belfast if I had to, maybe somewhere neutral, it doesn???t matter to me.
???It???s the biggest fight in the UK at the minute ??? Amir Khan v Kell Brook possibly ??? but this one has been brewing a lot longer. It???s a huge domestic showdown.”
Quigg hasn’t boxed since damaging is left hand back in November but is back in training and hoping to fight again round the end of May.
He flew out to Belfast and posed for photos with Frampton after his fight and was adamant that location disputes should not scupper their anticipated unfication showdown
“This fight is big not because of him and not because of me. It???s the full package. The British public want the fight.
???He draws the crowd in Ireland, we draw the crowd in Britain. It doesn???t matter where the fight ??? wherever holds the most, put the fight on.???