Not just budget but simple affordability. The cost of signing him, both transfer fee and wages will be huge, there's only a handful of clubs with the revenues (or outside investment) required. Arsenal's revenue last season would have been circa £380m, this season it might be £420-430m, that's roughly £200m less than the likes of Liverpool and Utd. Arsenal (and Spurs) have funded their transfers in the last couple of seasons with debt, that cannot continue and will have to start being paid at some point, they're not realistically going to be able to fund a £100m+ transfer and pay the huge wages required to sign Bellingham.