Sallói worked his way through the youth ranks of Nemzeti Bajnokság I club Újpest FC in his home town of Budapest, Hungary beginning in 2005.

[4] In 2014, he traveled to the United States as a foreign exchange student and attended Blue Valley Northwest High School in Overland Park, Kansas.

At that time he joined the U18 academy of Sporting Kansas City of Major League Soccer coached by fellow Hungarian and friend of his father István Urbányi.

[7] On 13 January 2016, it was announced that Sallói had returned to Sporting Kansas City, signing a professional, first-team contract and becoming the club's fourth all-time homegrown player.

[5] Prior to the 2016 Major League Soccer season, Sallói was loaned to Sporting's reserve side, Swope Park Rangers.

[17] The team won the bronze medal in the tournament after beating the Czech Republic 5–4 on penalties in the 3rd place match in which Sallói came on as a 46th-minute substitute.

[5] He made his debut for the Hungary national football team on 2 September 2021 in a World Cup qualifier against England, a 0–4 home loss.

[5] His father István had been a teammate of Sporting Kansas City manager Peter Vermes at Győri ETO FC.