The argument generally is: He lived in Portugal. He was married with a portuguese woman. All the proof he was Italian or genoese is said to be very speculative.
The consensus is actually the opposite. The overwhelming amount of evidence shows he is Italian, from Genoa (or around Genoa), while him being from Portugal, or from Spain as some say, are fringe unsubstantiated theories.
As proof of his Genoese origins, we have word from the man himself, as Columbus in a copy of his last will dating from 1498 says for Genoa: "insomuch as I was born there and came from thence" (Source, page 198). A quick google search shows various conspiracy sites doubting the authenticity of said will, and as I am not invested enough to research this issue, let's continue with other evidence.
His own son, Ferdinand, writing a biography of his father, places his origin in Italy and Genoa. And while Ferdinand expresses some doubt and lack of assurance of where exactly he was born, all the candidates he lists are around Genoa (Source, page 4) and on another place says for Columbus "and knowing that many of his Genoese countrymen lived in that city" (Source, page 14) and in general writes as if he was from Genoa.
Other sources also exist. I have at hand book Christopher Columbus and the Enterprise of the Indies which collects and translate various primary sources on Columbus, and in it we can find a legal document from Genoa in 1470 mentioning one Domenic Columbus and his son Cristopher. There is also a famous letter from Michael de Cuneo, a Genoese who accompanied Columbus on his second voyage and who pretty openly calls him a Genoese man. There is other less direct evidence, like frequent contacts with Genoa. Plentiful letters Columbus sent to Genoa (example) or to an Genoese ambassador at Spain (example) and others. Early chroniclers, like Spaniard Oviedo, writing in 1530-1540s identified Columbus as coming from Liguria, Italy (source) and Portuguese chronicler Barros in his 1550s work Décadas da Ásia ("Decades of Asia"), first decade, third book, chapter XI, also says Columbus came from Genoa (Source in Portuguese) and so on and on.
On the one hand, we have documented words from the man himself (if authenticity is accepted ), his own son, his contemporaries, and chroniclers immediately following.
On the other hand, we have as evidence: him living in Portugal and marrying a local woman? That's not even a case, really