The duo were caught up in a carjacking attempt last month by a gang on mopeds. According to the Times, both players now said to be under 24-hour security.
Two men were arrested after appearing at Ozil's north London home on Thursday and neither player was included in Arsenal's matchday squad for their 1-0 win against Newcastle in their Premier League opener on Sunday.
The Times are reporting on Monday that "eastern European gangsters are believed to have warned against any future attacks on the pair,” leading to an escalation of threats by a rival gang.
Sead Kolasinac fought off masked attackers who attacked him and Mesut Ozil
Image credit: Eurosport
The Sun have quoted a source saying: "Word came back from the other gangsters that they would not be intimidated.
"To show they were serious, the thugs issued more threats to the Arsenal players and refused to be cowed. It was thought that originally the robbers wanted their watches but now they have been told that they want everything the pair have."
Arsenal boss Unai Emery said he is unsure when Ozil and Kolasinac will return to the squad with their safety still a concern.
"The club is managing that situation. We want to be available with them," he told reporters.