ROME, Nov 10 (Reuters) - Telecom Italia's (TIM) CEO said on Friday he was available for a second term to oversee the reshaping of the phone company after the sale of its domestic fixed-line grid to U.S. fund KKR.

Asked where he saw himself in spring 2024, when the board's mandate will end, Pietro Labriola told Class-CNBC television that "having kicked off this project, I feel the burden and the responsibility to complete it".

"But it's not up to me to choose. I give my availability (for the job)," Labriola, who was appointed as CEO in January last year, added.

TIM's board agreed on Sunday to sell its prized landline grid to KKR in a 19 billion euro ($20.28 billion) deal, a key plank of Labriola's plan to revive the debt-laden group.

The sale, a first for a former phone monopoly in a major European country, is backed by the conservative government led by Giorgia Meloni but was heavily criticised by TIM's top shareholder Vivendi.

The French group has threatened a legal challenge to the decision.

Labriola called for greater dialogue among TIM's shareholders to reach a solution, as the sale of the network "redefines the industrial strategy of the group and has an impact on the telecoms strategy of the country".

"Everyone has the right to give indications... but we need to open up more and more to dialogue and avoid transforming everything into a confrontation," he said, when asked whether relations with Vivendi were "cordial".

Labriola added that as CEO he was available to discuss all positions, and that he also had to respect the opinions of all shareholders and not only some.

($1 = 0.9370 euros) (Reporting by Elvira Pollina, writing by Giulia Segreti, editing by Gianluca Semeraro)