He said he was informed of this personally by Lee on Tuesday, channelnews.asia.com reported.
Zainul said the prospect came as a surprise and he thanked the prime minister for the confidence.
He said he is open to taking up the post, adding that "it is difficult to say no".
Still, Zainul stressed he wants to focus on winning the General Election first - as part of the People's Action Party (PAP) team for Aljunied GRC.
Speaking to MediaCorp, he said "I thanked PM for his confidence. But I still need to win this election. And I am looking forward to getting the support of the people, before I can even be elected as Speaker."
He added: "It came as a surprise to me, because I was focusing on the elections, and I was also looking forward to attending unfinished business in the Ministry since I am handling relations in Southeast Asia, Middle East and South Africa, and I wanted to continue on building the good relations."
On what he thinks of the role, Zainul said from his experience in Parliament, the Speaker can play a difficult yet sometimes challenging role in trying to be fair but firm.
The last Speaker of Parliament, Abdullah Tarmugi, has retired from politics.