UPDATE 2-Onex-Greenbriar group preferred ILFC bidder-source

* Consortium wants to keep ILFC intact -source

* No exclusivity agreement with bidder -source

* AIG shares up 4.3 percent (Adds background, share price, byline)

By Paritosh Bansal

NEW YORK, June 5 (Reuters) - American International Group Inc (AIG.N) has picked a group that includes private equity firms Onex Corp (OCX.TO) and Greenbriar Equity Group as the preferred bidder for its aircraft leasing unit, a source familiar with the situation said on Friday.

However, the insurer has not signed an exclusivity agreement with the group for International Lease Finance Corp (ILFC), the source said.

AIG declined to comment. Onex and Greenbriar were not immediately available for comment. The source asked for anonymity because the talks are private.

Onex manages about $10 billion in total, while Greenbriar manages about $1.5 billion and focuses on investments in transportation.

The consortium wants to keep the aircraft leasing business intact, the source added.