The Wales full-back was named player of the series as the Lions beat Australia 2-1, and Blues chief executive Richard Holland labelled him "the only candidate" for the world player of the year award.
But with only a year left on the 24-year-old's contract at the Arms Park, Holland could face a battle to hold on to his prize asset.
Holland told the Independent: "Whether or not Leigh will stay I don't know but he's a Welsh lad and we're hoping he will commit to the Blues.
"We have already had discussions and we will be working hard with Leigh to lock him into the Blues for the foreseeable future.
"We are extremely proud of the achievements of all our Lions. Leigh must be the only candidate for the IRB's World Player of the Year after destroying Australia.
"Now we are going to do everything in our power to secure his services. We're trying to put together, within our means, a package which will enable Leigh to stay at the Arms Park and in Welsh rugby."