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You can make an offer of 80-20 then do financing with 80-10-10.

Once they accept the offer all they care about is getting their money. Note you still Need to show in your offer letter that you have the cash to do 20% down.

Note an 80-10-10 is both an 80-20 mortgage along with a 10% loan.

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Sure you can even do 100% cash and then refinance to 80-10-10. Although in both cases the rate is usually 50 bps higher, and all refinances essentially have a 1 point origination fee (this is often "baked in" so many people think it doesn't exist).

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