Private-equity firm TPG has agreed to buy cable-television providers RCN and Grande Communications for about $2.25 billion including debt, according to people familiar with the matter, in a bet that demand for new broadband services will fuel growth in the businesses.

Google Capital, Alphabet Inc.’s growth-equity investment fund, is taking a minority stake in the companies, which are currently owned by private-equity firm Abry Partners, some of the people said. The transactions could be announced Monday, they said.

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