Is it permissible to give zakat to a family whose financial situation is weak and has an income, but is going through financial hardship and needs money for medical treatment, knowing that if they take a loan, they can repay the amount later?

The poor and the needy who are eligible to receive Zakat are described in Fatwa no. 103013. If the description applies to those mentioned in the question, then it is permissible to give them Zakat. Their ability to borrow and repay debts later does not affect this, as the determining factor is their current state.