Is it permissible to request to rent a shop whose lease agreement is about to expire, knowing that the requester is aware of the current tenant's intention to renew his contract?

If the tenant intends to renew the contract, and there is a common practice and custom of renewal, then it is apparent that bidding over another's bid is prohibited. Al-Mughni by Ibn Qudamah states that bidding over another's bid is forbidden if the seller explicitly declares satisfaction with the sale, which is what the prohibition addresses. However, if the seller does not explicitly declare satisfaction or shows indications of dissatisfaction, then bidding is not forbidden, and sale by auction is permissible. If satisfaction is apparent without an explicit declaration, the default is permissibility of bidding, but some scholars tend to deem it forbidden.