I would stick with the past tense. I texted Tim this morning just to see how he was doing. Whether he has replied or not is irrelevant.
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fivejedjonBackshifting (present tense to past) is always correct. Retaining the original tense is acceptable if the situation referred to still applies., so 'to see how he is doing' is possible in your sentence.What do you mean by still applies?
fivejedjonIf, at the time of reporting the texting, you are still interested in knowing how he is doing, then backshifting is not obligatory.Thanks again!