Britain last Friday announced that it would accelerate its COVID-19 vaccination programme to try to contain a fast-spreading variant first identified in India, which could knock a reopening of the economy off track.
"We'll make a final decision for step four, which is the biggest step on the roadmap. We'll make that final decision on the 14th of June," Hancock told Sky News.
The last of four stages in the removal of lockdown restrictions is currently scheduled for June 21 and could ditch social distancing measures entirely.
The third stage is due to start on Monday.
Among other measures, the easing will allow people to hug each other and enable pubs and restaurants to serve customers inside.
Hancock also said that early data showed that vaccines were effective against the Indian variant, Reuters reported.
"There is new, very early data out from Oxford University, and I would stress that this is from the labs, it's not clinical data, and it is very early, but it does give us a degree of confidence that the vaccines work against this Indian variant," Hancock said.