Mr Eustice told MPs the Government was "already working with the hospitality and pub sector to identify what social distancing measures they might be able to put in place to make that work properly".
His comments during Commons environment questions came as Tory David Johnston (Wantage) said pubs operating as takeaways "is not a substitute for being open properly".
Mr Eustice said: "As the Prime Minister has outlined, we intend that the hospitality sector, including pubs, would be able to tentatively start gradually opening hopefully during the month of July - subject to the epidemiology supporting such a move."
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Mr Eustice also said it is estimated that only a third of the usual European fruit-picking workforce will be available to work in the UK.
Shadow environment minister Daniel Zeichner said: "The foolish dismantling of the seasonal agricultural workers scheme, now made worse by the Covid crisis, means we're facing an alarming shortfall in the experienced 70,000 people needed to pick our crops."
Mr Eustice replied: "We estimate that probably only about a third of the usual East European workforce that would come to work on our farms is either here or in some cases has continued to come.
"That means that this year we will need a British workforce to step up and assist in getting the harvest in this year and we're very encouraged by results so far."
Shadow environment, food and rural affairs secretary Luke Pollard pressed the Government to publish an "urgently needed" food strategy with a focus on coronavirus.
He told the Commons: "Food bank demand is surging, up on average between 60% and 80% from pre-virus level. Now unemployment is soaring by nearly 70% last month, it is clear we need measures that match the scale of the crisis."
Mr Eustice responded: "The food strategy that is being developed by Henry Dimbleby, one of our non-executive directors, will do an initial report setting out the approach and the nature of the challenges in the autumn this year and the final report is expected in the early part of next year."
Mr Eustice insisted the Government will protect the UK's food standards in all trade negotiations in response to concerns from the SNP's environment spokeswoman Deidre Brock.
Mr Eustice said: "This Government's got a very clear manifesto commitment that we will protect our food standards in all of our trade negotiations.
"Certain practices such as chlorine washes on chicken or hormones in beef are subject to a prohibition on sale in the UK and that law remains in place.
"And it is also the case that as we advance trade negotiations will all third countries, animal welfare is one of the issues that we will be seeking to promote."
Shadow minister for natural environment and air quality Lloyd Russell-Moyle (Brighton, Kemptown) said the Government had "dismissed" putting air quality targets on the face of the Environment Bill, adding "but Covid surely has changed all that".
He called on the Government to "agree a form of wording which will require ministers to set targets on air quality to reach the WHO standard by 2030 and help save British lives".