The Toffees signed the 35-year-old goalkeeper last summer and he has been the number 2 to Jordan Pickford at Goodison Park.
He is in the final year of his contract and his future at Everton has come under the scanner due to interest from Manchester United.
Erik ten Hag wants to bring in a strong number 2 to David de Gea and Begovic has emerged as a target for the Red Devils.
However, Everton do not want to sell and Lampard is determined to hold on to the veteran goalkeeper.
The Everton boss values his experience and the ability to step in between the sticks whenever needed.
He is also considered a homegrown player and helps Everton with their 25-man squad that they have to register with the Premier League after the window closes.
Any move away from Everton would have to depend on the club bringing in a replacement for him.
Manchester United are interested in getting their hands on Begovic but Everton are confident that they can convince him to stay.