* Russian non-state oil companies support cuts extension
* Oil prices under pressure as markets still awash with fuel
* OPEC favours cuts extension, counts on Russia - sources
By Vladimir Soldatkin and Oksana Kobzeva
MOSCOW, March 24 (Reuters) - Russia's private oil producers are ditching their scepticism and lining up behind an extension of a global deal to cut output after previous oil price increases compensated for lost income.
Oil prices have risen since November when the Organization of the Petroleum Exporting Countries (OPEC) reached its first deal to lower output since 2008, with some non-OPEC producers, including Russia, joining the move.
A decision on whether Russia backs an extension will ultimately be taken in the Kremlin, and the private oil companies have limited say. But their change of heart is a tentative sign that oil price rises have diluted some of the opposition to global production cuts.
As oil prices are still under pressure, OPEC sources told Reuters this week that Russia and other non-members need to remain part of the initiative to prolong the deal beyond June.
Russian state oil companies Rosneft and Gazprom Neft along with private Lukoil account for the bulk of Russia's cuts. Russia had cut output by 160,000 bpd by mid-March out of the targeted 300,000 bpd, now expected by the end of April.
Vagit Alekperov, chief executive of Russia's second-largest oil producer Lukoil said last week it was "expedient" to continue cuts, as the deal has already brought its first results.
Alekperov's comments represent a U-turn from last year when he said there was no point in Russia doing any deal with OPEC.
Another Russian major Tatneft, which is controlled by the Russian region of Tatarstan, said it stood ready for output reduction.
"We are ready to cut production at the levels, which lead to a financial result... but without harm to future output," the company said in emailed comments to Reuters.
Mid-sized company Russneft said that the company was ready to extend production cuts if it "serves Russian interests." "We hope the oil prices rise will offset out losses from the production cuts," Russneft said in emailed comments.
Russian Energy Minister Alexander Novak has said it was premature to decide for now if the deal should be extended, adding it would be clearer in April and May, closer to the deadline.
The energy ministry declined to comment. Kremlin spokesman Dmitry Peskov did not reply to a Reuters request seeking a comment.
Russian state oil companies, Rosneft and Gazprom Neft, were initially reluctant to cut as both planned to increase output. Both have complied with the cut following the order from President Vladimir Putin. (additional reporting by Olesya Astakhova; editing by Katya Golubkova and Keith Weir)