You do not say whether the value of the replacement contract takes you above the EU Procurement limits (currently £113,057 if you are in Central Government or £173,934 if you are in other parts of the Public Sector), for the life of the contract.
If the cost is below those limits, you are bound to comply with your organisation’s Standing Financial Orders and Instructions for tenders and the Government’s Transparency Agenda by publishing details of contract opportunities above £10,000 on the Contracts Finder database. I shall assume the value is above the EU limits.
You also do not say whether this is for goods or a service? I shall assume the latter in this response but the same principles will apply to goods.
From what you say about the contract, you are not entitled simply to extend again, whether by “negotiation” (not strictly permitted) or by simple extension. You need to go through an appropriate EU tendering exercise. Failure to do so would run the risk of challenge from other interested suppliers and from External or Internal Auditors as well as, most importantly, being contrary to getting best value for money for the Tax Payer from a market which is different from the one which applied when the initial contract was let.
If you do not have enough time left before the current extension expires to conduct a tender and award a contract, how serious would be the lack of the service for the time it takes to go through the tender/award cycle? If it is not serious, best to get on with it even if that would mean a gap in the service. But that possibly begs questions as to why it needs to be replaced at all. Could it be a savings candidate?
Assuming the service is essential and you do not have time to go through the procurement cycle without incurring a service gap, you may after all have to seek an extension to the contract but this time a “qualified” extension in the context of a documented procurement strategy to replace the contract. That should be enough to satisfy the main likely objectors - those External or Internal Auditors or other contractors who want the chance to deliver the service.
Beware of extending for another year, only to find it is unnecessarily long (wasteful) or too short (embarrassing, time consuming and weak to have to extend again). The Procurement Strategy needs to be realistic.
By the way, if you did not have an exit strategy in the original contract and you need information from the current contract to be available to help a new contractor hit the ground running, that would be the opportunity to “negotiate” that with the current contractor. But negotiating that is another topic all together.
One final though: how unique is your service? Are there in existence Framework Contracts let by other parts of the Public Sector to which you could have access? Or are other parts of the Public Sector obtaining these services from some Buying Consortium? They could meet your needs, possibly at a better price through leverage of larger volume purchasing or at least without a gap in service? Have a look at the (former) OGC/Buying Services web page and do an Internet search for Buying Consortia for contacts to check. You may even find other parts of the Public Sector local to you have the same service. Ask a few suppliers if there is no other way of finding out