(Adds context) MILAN, April 28 (Reuters) - Telecom Italia...

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MILAN, April 28 (Reuters) - Telecom Italia (TIM)'s TLIT.MI CEO Pietro Labriola is expected to propose to directors to seek further improvements of multi-billion bids for the company's landline grid at a board meeting on May 4, two sources familiar with the matter said on Friday.

The sale of TIM's landline grid and submarine cable unit Sparkle is a key plank of Labriola's s plan to slash the group's 25 billion euro ($27.6 billion) debt and revive the former phone monopoly's domestic business, under pressure for years due to stiff price competition.

Rival suitors KKR KKR.N and a consortium comprising Italian state lender CDP and Macquarie (MQG) have offered 21 billion euros and 19.3 billion euros respectively, sources have said, each raising their initial offers by 1 billion euros.

Both bids did not come close to the 31 billion euros requested for the asset by TIM's top investor Vivendi VIV.PA , and also fell short of TIM's expectations, sources have said.

French media giant Vivendi, which owns a 23.8% stake in TIM, in recent weeks has been piling pressure on Labriola and other managers to draw a line under the ongoing bidding process, given a too large valuation gap and concerns over the sustainability of TIM's remaining service business.

($1 = 0.9056 euros)