Lots of helpful suggestions, but no-one has yet answered the specific question of what is a reasonable price for this journey.
£105 sounds surprisingly cheap for such a long distance taxi from London.
Popular London taxi firm Addison Lee quote around £250 -
they're not the cheapest, but their prices are usually pretty competitive.
For a regular taxi firm, £250 isn't surprising to me at all (I was expecting around £230).
Crowdsourced taxi site/
app Uber , which is usually the cheapest option, quotes £104-£139 for their smallest, cheapest cars
[Uber won't actually book inter-city journeys, as Ankur Banerjee points out, but their quotes give an indication of plausible prices].
£105 is cheaper than three people taking an off-peak train ( £39.50 each ), and London-based taxis are notoriously expensive - it can easily cost £50 just to get from one side of London zone 2 to the other.
Long-distance taxis are particularly expensive since you're not just paying for your journey, but also for the (usually empty) cab driving back to base after dropping you off.
As well as the driver's time this includes double the fuel (which is expensive in the UK) and all the business they missed out on while driving back to base.
If they get stuck in traffic, if you don't show up, if you're very late or if you do a runner, it's much more disruptive than for a short journey.
Expect them to want at least part of the payment up front.
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Your best bet would be to contact Southampton based firms directly.
Local taxi firms include Radio Taxis and West Quay Cars .
I've used both for local journeys in and around Southampton.
When I've been on trips abroad with work we've always used English Rose to get to/from the airport, and have been very impressed.
Even when we arrived 8 hours late at 5am
they were there waiting for us.
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For long distance taxi hire within the UK, there isn't usually a published rate .
A lot of private hire and taxi sites exist (none very user friendly) and a lot of what you actually pay in the end depends on what you can negotiate with a driver - because a lot of them have standard rates on paper, but say if you have a return journey or repeat customer, they'll be happy to negotiate lower fares.
That said, like Gagravarr commented, a train will be cheaper, quicker, and less hassle.
travel.stackexchange.com/users/108/Ankur Banerjee
we will have a lot of luggage and
a service door to door is what I am looking for Define "a lot of luggage".
The referenced website has a Minibus, 8 people / 8 large cases for 50% more so you are now looking at £150
+.
It's a long trip, and "a lot of luggage" in England isn't going to move cheaply.
You could try a delivery service like DHL, assuming they have a counter at the cruise terminal.
Otherwise, it's going to be rather expensive.
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Checker Cars, based at Southampton Airport, charge about £105 Heathrow to Southampton, and about £140 central London to Southampton, any time, any day.
They operate some Ford Galaxies and Vauxhall Zafira Tourers, both of which are luggage-eaters.
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I am looking for the best option and I found a website  https://www.londondrivers.com/southampton-london/  and the price is 105 GBP ... Is this price correct?    EDIT ADDED from comment posted as answer:    We need a taxi from the Ocean Cruise Terminal because we will have a lot of luggage and a service door to door is what I am looking for
How much does a taxi from Southampton to London cost?
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travel.stackexchange.com/users/27640/Mike
£105 sounds surprisingly cheap  for such a long distance taxi from London, and this is cheaper than three people taking an off-peak train (£39.50 each). There usually isn't a published rate for long distance travel within the UK, but you could contact Southampton based firms directly and try to negotiate a good price. Published rates for long distance taxi rides in the UK do not generally exist. Taxi forms give a quote for the trip and from Southampton to London the range is 104 pounds to 250 pounds. Local taxi companies probably giva a better rate. Negotiating the rate with the driver from a private taxi firm is often the norm.
105 pounds for a taxi from Southampton to London seems very cheap as it is cheaper than the train for three people. The price for a taxi from Southampton to London would appear to be between 104 and 250 pounds. The best idea is to contact local taxi companies and Checker Cars quotes 105 pounds to Heathrow and 140 to central London. Long distance taxi rides in the UK are often negotiated between the driver and the passenger as there is not usually a published rate.