Lord Sainsbury of Turnville and Lord Sainsbury of Preston Candover sent the letter to Sheikh Hamad bin Jassim bin Jabr al-Thani last week, according to reports in The Times.
In the letter, the family said they would not back a takeover that left the company in too much debt or sold off its property portfolio, the reports added.
A source also told the paper that the family had not been contacted by Qatari-based investment fund Delta Two since it made its offer last week.
The family has also written directly to Delta Two asking for more information on its approach, according to reports in the Daily Mail.
Delta Two met with Sainsbury's chairman Philip Hampton yesterday to persuade him to open Sainsbury's books for due diligence but the talks were inconclusive, the paper said.