Russia's consortium AlfaAccessRenova (AAR), a co-owner of TNK-BP, is demanding that Dudley step down accusing him of putting the interests of British Petroleum (BP) first.
Speaking at a news conference in Moscow on Saturday, Dudley, who headed the joint venture for five years, said he was determined to remain in the company's management and lead the company to success.
Founded in 2003, the Russian-British venture TNK-BP, Russia's third-largest oil producer, is jointly owned by BP and AAR. The consortium, representing Russian investors Alfa Group, Access Industries and Renova (AAR), holds 50% in the energy company.
Dudley was appointed president and CEO in 2003, when the JV was set up.
He said on Saturday that his main priority as the CEO at present was to ensure successful production and financial performance of the company.
Russian shareholders in TNK-BP want to expand into Turkmenistan, Kazakhstan and Venezuela, where TNK-BP has a competitive advantage. BP has resisted the idea, as well as plans to take part in high-risk projects in countries including Iraq, Uganda, Syria and Libya.
Earlier this week business daily Kommersant said the Russian shareholders were seeking to replace Dudley with Russian tycoon and TNK-BP's executive director Viktor Vekselberg.
The Russian-British oil venture has come under intense pressure from the Russian authorities this year. In March the Federal Security Service raided the company's headquarters and the Moscow office of BP, and carried out new searches at BP last week. In March a TNK-BP employee was arrested on suspicion of industrial espionage.
British shareholders at TNK-BP reiterated their trust in and backing for Dudley on Thursday during informal talks with Russian shareholders in Cyprus.
Analysts say the ongoing dispute between Russian and British shareholders could lead to the company being bought up by a Russian state giant, most likely Gazprom but possibly Rosneft.