SueDonim
Are you in England? In Scotland in that situation a ‘closing date’ would be set, and everyone would make their best offer by that date. The seller can accept any offer, it needn’t be the highest eg someone might be a cash buyer.
Is there a method of doing that in the English system?
It happens a lot round here. If several people are interested the Estate Agents asks for each party's "Best and Final" offer. The seller is given all the details: offer level, cash or mortgage, the buyer's own sale position and so on so that the seller can make a sensible decision based on all the information.
We were involved in this situation when buying for my daughter and found that the EA was not wholly impartial, recommending someone as a "cash buyer" when they had not, in fact, exchanged on the sale of their own property. The buyer accepted their offer but it subsequently fell through when the seller realised that they could not exchange contracts as their situation was not as stated.
The seller came back to us and we exchanged on the sale with them.
Sometimes the EA is keen for you to accept a particular buyer because they are also acting for them on their own sale. Or sometimes they ask for "Best and Final" offers simply in an attempt to push up the price even if there is no real competition.