pensions contributions based on savings interest?

qwakuk is absolutely right.

But some more detail might be useful and I am at a loose end :-)

You can contribute however much you like, but there is a limit on the amount you can contribute and gain income tax relief.  That is limited to your "net relevant earnings", and if your grossed-up contributions exceed that value, you will have to declare it and cough up the excess relief granted.  NRE is not quite the same the "Pay, pensions, profit etc." section on your self assessment tax calculation which is taxed at the earnings rates: notably, pensions in payment are taxed as earnings, but don't count towards NRE.  Neither it the same as "Total income on which tax has been charged" value, even if all your income is earnings, because that value is after the deduction of allowances.  And neither is it the same as the amount of annual allowance you might have: even if you carry forward say £40k AA from previous years, you still can't make that contribution and gain tax relief on it unless you have the NRE to count it against. 

The upshot is that when people ask your question, what many of them actually mean is "I've got £10k in savings income and I earn £30k, can I put the £10k interest in my pension and get tax relief?"  And the answer is often "Stop compartmentalising.  Unless you are already whacking almost your whole salary into it, yes you can contribute another £10k gross, because it's more salary you're putting in, you're keeping the savings income to buy food with".  Furthermore, it doesn't matter if you pay tax on the earnings already or not.  You can contribute the *whole* NRE and get the tax relief, including the bottom 12k+ which is covered by your allowance(s) and on which you were never going to pay tax in the first place (!) 

Also: salary sacrifice, NI, inheritance tax avoidance, maximisation of 5k zero-rate savings band, minimisation of marginal effects child benefit, interaction of pension relief with gift aid VCTs seafarer's deduction etc. etc. 

But if your earned income is zero or thereabouts. It's easy.  See qwakuk 

For simplicity I'll assume you are talking about a standalone SIPP outside of work.  Contributions are made net then the scheme applies "relief at source" by claiming the tax back from HMRC.  Your £30k earner pays (again, round numbers for simplicity) 20% on £20k=£4k tax, having used his/her personal allowance on £10k of the earnings plus a couple of £1000 rental income or something which we will disregard.  Now he has £26k and has paid £4k tax. 

If he puts £16k of that in the SIPP, the scheme will claim £4k back from HMRC.  The total, gross contribution is £20k.  He is back in the position he would have been if the tax had not been paid. 

If he puts £24k of that in the SIPP, the scheme will claim £6k back from HMRC.  The total, gross contribution is £30k.  This is the maximum contribution that can attract tax relief, as the NRE is also £30k.  And yes, he still has £2k of his earnings left over because he's getting tax back on the £10k that was covered by the allowance. 

If he puts other money in too, say £40k total, it depends how the individual scheme handles it.  In my SIPPs, when I make a contribution I am asked what my NRE is, and (I presume that) the scheme would therefore do something about it if the NRE wasn't enough: either claim less tax back, or more likely refuse the payment to avoid having to deal with it.  I'd have to lie or be mistaken about my likely earnings over the remainder of the tax year in order to just plough the money in regardless.  If that happened, ultimately the obligation to pay the right tax remains with the taxpayer: I'd call HMRC, they'd [guessing here] eventually figure out how to deal with it and if the scheme had done what it was supposed to given the numbers I'd provided to it, I'd (rightly) be the one getting the balancing bill to put things back as they should be. 

Note that hardly anyone does this.  You're on the hook for tax if you ever take the money out, so the only reason to make a contribution without income tax relief is for estate planning, which an incoming Labour government may not be very keen on.  It becomes relevant for third party contributions e.g. an employer. 

There's no *additional* cost to paying in unlimited sums, merely a requirement to pay the tax as normal rather than the excess contribution being allowed excess relief.  It's your money and once you've paid the income tax via PAYE or self assessment you can put it into a pension as contributions in excess or NRE or you can put it in a hole in the ground.  I'd rather have it in a hole in the ground a.k.a. an ISA, where at least I can then remove it when I retire without being stung for tax on removal.  The sums *below* the NRE, however, I'd rather have in a pension for the immediate relief plus other tax planning benefits, on the assumption that the eventual taxes on withdrawal will be less. 

Those gov.uk pages are fine for an overview but are lacking in detail in order to make things "simple" but if you want a more detailed, less ambiguous resource try the one HMRC use: www.gov.uk/hmrc-internal-manuals/pensions-tax-manual/ptm040000

If you only have savings, you don't need to worry about it.  £3600 gross, £2880 net.