The Scot has scored 20 goals in 20 appearances for the Swedish leaders this season, form that has alerted several clubs across the Premier League, Serie A, Ligue 1 and Turkey's Süper Lig, with Manchester City and Chelsea both credited with interest according to reports in Sweden.
Sevilla's opening game of the new campaign against Rayo Vallecano is just ten days away, and sporting director José Ignacio Navarro has made signing a new centre-forward one of his top priorities, sending a formal proposal to Sirius as talks continue over personal terms and a fee.
Ure, 22, is valued at around €5m, having joined Sirius from Anderlecht for a reported €750,000 fee. He began his career at Rangers, making just two senior appearances before rejecting a contract extension, and a similarly low-key spell in Anderlecht's academy preceded his move to Swedish football.
"It's normal that, when you're young and performing well in a good league, interest comes from big leagues and big clubs," Ure told BBC Sport recently. "It's something that will interest me if I feel it's the right step for me. But we have to wait and see. The summer transfer window is long."
Sevilla are understood to be working on as many as five incoming signings this summer, with talks over Girona midfielder Giorgi Kochorashvili also intensifying, as Navarro looks to strengthen the squad before the market closes.
Ure's rise has been rapid since his move to Sweden: after joining on a four-year deal, he scored 16 goals in 35 appearances across all competitions in his first season, and this year's return has put him second only to Bayern Munich's Harry Kane among the goalscoring charts in Europe's top 15 leagues, ahead of established forwards such as Luis Suárez and Vedat Muriqi.
He also has interest from clubs in Italy and France, including Milan, Napoli, Monaco and Lille, while Turkish sides Galatasaray, Fenerbahçe and Beşiktaş have also been credited with tracking his situation, according to Swedish outlet Fotbolltransfers.
Sirius sit clear at the top of the Allsvenskan table and are chasing a first league title in their history, meaning any departure for Ure is more likely to be resolved before the season restarts than during it.