I have read of some people who got both vaccines at the same time, and their experience was not good.
However perhaps a different kind of Influenza vaccine combined with a COVID-19 vaccine might make things easier.?
This approach is now being tested.
Pfizer and BioNTech Announce Positive Topline Data for mRNA-based Combination Vaccine Program Against Influenza and COVID-19.
Lead formulations evaluated in the Phase 1/2 study demonstrated robust immune responses to influenza A, influenza B, and SARS-CoV-2 strains
Safety profile of the mRNA-based combination vaccine candidates consistent with the companies’ COVID-19 vaccine
The companies plan to start a pivotal Phase 3 trial in the coming months.
In this clinical trial, the vaccine candidates were compared to a licensed influenza vaccine and the companies’ Omicron BA.4/BA.5-adapted bivalent COVID-19 vaccine given at the same visit.
The data from the trial showed that the companies' lead formulations demonstrated robust immune responses to influenza A, influenza B, and SARS-CoV-2 strains.
pfizer.com/news/press-relea...
Dick