Since you want this answered as close to the referendum date as possible, I did find one survey by What UK Thinks from the autumn of 2016 (the Referendum was held in July). It is pretty elaborate, so I suggest you read it all for the details, but its summary findings were:
Nine in ten people would like free trade with EU countries to continue.
But at the same time, as many as seven in ten (70%) think the UK should be able to limit the number of people from the EU who come here to live and work, including 55% of people who voted
to remain in the EU.
The public is split [49% vs 51% overall] over whether Britain should accept freedom of movement of people in exchange for free trade with EU countries.
This split is even clearer when we look at those who voted to remain in the EU and those who voted to leave. [30-70 in the Leave vote vs. 70-29 in the Remain.]
Frankly this is not much of a Brexit model; more like a "why do you [not] want Brexit" broling down to the tradeoff between immigration and free Single Market access.
But this survey does find a rather prevalent "have the cake and eat it" attitude among the British public at this time:
In short, the apparently widespread wish amongst the British public to remain part of the single
market in goods and services but to opt out of the obligations the EU’s freedom of movement
provisions is not confined to those who voted to Leave. It is also to be found amongst many who
voted to Remain. Such a deal would, it seems, be backed by a majority of voters on both sides
of the referendum.
But, as we have already noted, many political leaders in the EU have indicated that this is one
deal that will not be on offer to the UK when the negotiations take place.
There's also a RAND study from February 2017 (which also has a later, longitudinal follow up.) This RAND study is actually synthesizing a Norway-like model of Brexit based on the responses. The conclusions of the first RAND study were that:
The British public want a deal.
The British public want a deal on Brexit and are willing to compromise to get one. Netting out the positives and negatives, we find that the current situation of EU membership is worth about £14 per household per week more than leaving the EU with no deal.
People are more concerned with managing demand for public services than simply restricting freedom of movement.
This is particularly true of those who voted to leave the EU.
People highly value having access to EU markets for trade in goods and services, but also would like the UK to be able to make its own trade deals.
The British public place the greatest value on having the ability to make trade deals and retaining access to the Single Market for trade of goods and services after Brexit, more so than restricting freedom of movement, increased sovereignty and reduced EU contribution.
People value the Single Market access and the ability to make trade deals.
The British public value the UK being able to make its own laws, but not as much as they value Single Market access or the ability to make trade deals.
People with degrees hold stronger views.
Education level was the most important explanatory variable in quantifying people’s preferences. Overall, those with university degrees preferred closer ties to the EU. They were more positive about the value of freedom of movement for holidays and working and strongly disliked options with severe restrictions on freedom of movements, such as requiring a visa to travel to other European countries for holidays (and requiring other Europeans to have a visa to travel to the UK). They were also less sensitive to the level of EU contributions and held differing views about the importance of UK sovereignty over its laws, preferring options where the UK is subject to EU laws in environment, employment and trade.
People prefer a final agreement which is close to an agreement that is similar to the Norway model.
Given the importance of making trade deals and retaining access to the Single Market, we find that the public place a positive value on a relationship like Norway’s current relationship with the EU, allowing for free trade with other countries while remaining within the single market, and accepting freedom of movement and some loss of sovereignty.
Do note that things like the backstop etc. were not in the public discussion sphere back then, so they didn't get included in the questionnaire. A longer version of seemingly the same study was published in July 2017. The follow up that I already mentioned conducted about a year later. Found pretty much the same. Alas, the Irish border problem isn't addressed by a plain "Norway model" like the one outlined above.
For more recent surveys on the more concrete (draft) Withdrawal Agreement between former PM Theresa May and the EU, see Is there a survey of the public opinion whether no-deal is preferable to the November deal?