Adds share prices, oil price moves, Woodside, Exxon and state premier Cook's comments in paragraphs 2, 4, 5 and 7

By Roshan Thomas and Helen Clark

June 15 (Reuters) - Australia's Woodside Energy WDS.AX said on Monday it was not aware of any proposal involving U.S. major Exxon Mobil XOM.N and was not engaged in discussions regarding a potential transaction, responding to recent media speculation.

Woodside's shares fell 2.7% in early trade on the Australian stock exchange on Monday, as oil prices slid 4% to their lowest since March after the U.S. and Iran said they had reached an initial deal to end their war and resume traffic through the Strait of Hormuz.

Its secondary U.S.-listed shares had risen 8.35% on Friday after Bloomberg reported Exxon was considering Woodside as a potential takeover target as part of efforts to deepen its presence in liquefied natural gas and Asian markets.

"In response to recent media speculation, Woodside is not aware of any proposal and confirms it is not in discussions regarding a potential transaction with Exxon Mobil Corporation," the company said on Monday.

Exxon in Australia declined to comment.

Western Australia Premier Roger Cook said on Sunday the state government would oppose any takeover that involved relocating Woodside's headquarters from Perth, where it has been based since the 1990s.

"Overseas interests have sought to acquire Woodside before and the reasons it was previously defeated have not changed," Cook said in a statement to Reuters on Sunday.

Woodside is Australia's leading LNG exporter and recently increased its stake in the Browse project to about 42%, after pre-empting an Inpex 1605.T buy of PetroChina's 601857.SS 10.67% share of Australia's largest undeveloped gas resource.

While oil prices fell, Woodside's shares held up slightly better than its nearest rival in Australia, Santos STO.AX, which fell 4.3%.