Social distancing guidelines have been kept under review as lockdown has gradually eased, but low case numbers mean embracing family members and friends could be allowed from May 17.
It will be the first time people are permitted have physical contact with those from different households in more than a year.
Ministers will meet next week and are likely to agree to relaxing the restrictions, The Sun and Times report.
One Whitehall source said: “The modelling is on track for step three as planned.”
International travel and indoor dining is also expected to resume on May 17.