By Mauro Orru and Olivia Bugault 

Shares in Telecom Italia SpA jumped Wednesday after a report that KKR & Co. is considering increasing its bid to take the telecommunications company private.

The share move came just days after the U.S. investment firm first expressed interest in Telecom Italia through an offer with an equity value of roughly 10.79 billion euros ($12.14 billion).

At 1510 GMT, Telecom Italia shares were trading 16% higher at EUR0.50.

The Italian company said late Sunday that it had received a friendly and non-binding indication of interest from KKR aimed at delisting it, paving the way for a possible buyout offer at EUR0.505 a share--a 45% premium on Friday's closing price.

Its current share price is still below KKR's tentative offer, despite Wednesday's rise.

A person familiar with the deal told The Wall Street Journal that KKR is "totally and solely" focused on the offer that was announced Sunday.

Bloomberg reported late Tuesday that KKR was debating increasing its bid for Telecom Italia to around EUR0.70 or EUR0.80 a share to overcome potential resistance from Vivendi SE, the media conglomerate steered by the family of French billionaire Vincent Bollore that has amassed a 23.75% stake in Telecom Italia.

Vivendi believes KKR's offer doesn't reflect the value of the company, a spokesperson told The Wall Street Journal. The French company is a long-term investor in Telecom Italia, they added.

Opposition by Vivendi, which has built its stake in Telecom Italia at an average price of EUR1.07 a share, was to be expected, analysts at Jefferies said earlier this week.

"In our view, the critical point is whether the offer will be approved by the board. If so, we think there may be room for negotiation on the valuation," Equita Sim analysts said.

A spokesman for Telecom Italia said the company's board would meet on Friday.

If successful, KKR's bid would make Telecom Italia the latest in a line of European telecommunications companies to be taken private this year. Billionaire businessman Xavier Niel, who founded French telecommunications company Iliad in 1999, unveiled an offer to take it private in July, following the precedent set by fellow entrepreneur Patrick Drahi earlier this year when he delisted Altice Europe.

Write to Mauro Orru at mauro.orru@wsj.com; @MauroOrru94 and Olivia Bugault at olivia.bugault@wsj.com