Analysts said the company needs deeper cost cuts to boost profitability, and Livingston's background may help achieving this, but Verwaayen played down that scenario.
"Have you ever seen analysts not talking about cutting more costs? Let the analysts do their analysis, that's their job," he told CNBC Europe.
He will be succeeded at the retail business by Gavin Patterson, currently managing director of BT's consumer division.
"Ian was the board's unanimous choice to succeed Ben in June," BT Chairman Mike Rake said in a statement.
"He will continue to deliver on a proven strategy."
Analysts welcomed the move, saying that Livingston's track record was good.
But " we would note that whilst the new CEO appointment is generally in-line with consensus, the timing appears to be slightly ahead of what many had expected," Michael Kovacocy, European telecoms analyst at Daiwa Securities SMBC Europe, said in a market note.
Verwaayen said he had not been put under pressure to leave. "You should go when people say why do you go not why aren't you going. You need to go when people still think it's too early," he told CNBC Europe.
In 1984, BT became one of the world's first state telecom companies to be privatised and was one of a string of national firms to be floated under Conservative Prime Minister Margaret Thatcher.
For years it struggled as regulators imposed heavy restrictions in a bid to encourage competition at the same time as its traditional fixed-line business started to decline.
Verwaayen joined BT in 2002 from Lucent Technologies and helped to open up BT's network to rivals at the same time spearheading a drive into broadband and network IT services.
BT shares closed 1.5 percent higher on Tuesday.
-- Reuters contributed to this report.