Kabel Deutschland is Germany's leading cable provider, providing television, telephony and broadband services to about 8.5 million connected households in 13 of Germany's 16 federal states.
Vodafone said in a statement it would offer Kabel Deutschland shareholders €87 in cash for every share.
The offer included the purchase price of €84.50 per share plus the €2.50 dividend announced by Kabel Deutschland in February.
Kabel Deutschland shares had closed at €84.10 on Friday.
The management and supervisory boards of Kabel Deutschland "welcome the transaction, (and) intend to recommend Kabel Deutschland shareholders accept the offer," Vodafone said.
Kabel Deutschland would provide Vodafone "with an attractive platform for TV and fixed broadband in Germany and creates a leading integrated operator with pro forma revenues of approximately €11.5 billion," the statement said.
Vodafone's bid comes amid rising interest in cable assets Europe. Last week, Kabel Deutschland confirmed it had been approached about a potential tie-up by Liberty Global.
Liberty acquired another German cable operator, Kabel Baden-Wuerttemberg, two years ago.
In London, Vodafone shares were outperforming the overall market, adding 1.18 percent, while the FTSE index was up only 0.25 percent.
In Frankfurt, Kabel Deutschland shares added 1.70 percent to reach €85.53.