Can you delay a closing of a house for a month?

There is a house I'm considering buying. It's been on the market for 6 months. The price got lowered in January, but it's still available. My lease for my apartment isn't over until July 31st. My realtor told me if we didn't close until June, the first payment wouldn't be due until August. Would most banks/etc. Be willing to push the closing until June if they had a confirmed buyer? Or be willing to accept an offer that stated closing would be June 1st?
Added (1). not sure why my question says a month, wasn't thinking I guess. It's nearly the end of April now… Put the offer in… Usually a month turn around, right? Just looking for some advice on this situation =)

Read more: Can you delay a closing of a house for a month?