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* Offer could come as early as next week - source
* Indicative offer was 81 euros/shr - sources
* Kabel Deutschland rise to record 83.91 euros
* Vodafone declines to comment
FRANKFURT, June 14 (Reuters) - British mobile firm Vodafone is considering raising its offer for Kabel Deutschland after an initial 7.2 billion euro ($9.6 billion) bid was knocked back by Germany's biggest cable operator, one person close to the matter said.
"The offer could come as soon as next week," the person said on Friday.
Vodafone declined to comment.
Vodafone sent a letter to Kabel Deutschland last week announcing its interest and indicating a price of 81 euros per share, three people familiar with the matter have told Reuters.
Vodafone has confirmed that it made an offer to buy Kabel Deutschland but has not provided details.
The world's second-biggest mobile operator has long been looking for its most important European market to meet growing demand from customers for television, broadband, mobile and fixed-line services - so-called "quad play" - from one provider.
Any deal, which would be Vodafone's largest since 2007, could help it better compete against mobile operators, which are cutting prices, and against the pan-European cable group Liberty Global, which has been on an acquisition spree.
Vodafone had 32 million customers in Germany at the end of March, making it the biggest mobile operator in the country. Kabel Deutschland is Germany's largest cable operator with about 15 million of the 28 million homes passed by cable.
Kabel Deutschland shares were up 3.2 percent by 1320 GMT after hitting another all-time high at 83.91 euros. Vodafone shares were up 0.9 percent.