Manager Russell Slade has said Cardiff are looking at whether Juan Cala broke club discipline rules on social media.
Aston Villa boss Paul Lambert has defended his decision to send striker Darren Bent out on loan to Brighton.
Central defender Konrad, 24, signs from SV Eintracht Trier 05 having spent the last two seasons at the club.
Tankulic, 23, joins after successful spells with FSV Mainz 05 II and VfL Wolfsburg II.
Both players enjoyed a successful period on trial with the club and join Paul Hartley’s squad ahead of the 2014/15 season.
Dundee manager Hartley said: “They impressed when we were away in Hungary and in the Manchester City game showed what they can do.
“They are both at a good age and fit into the squad we are building. They are two good additions.”
The pair joined Dundee for their pre-season training camp in Hungary at the beginning of July, and returned to Scotland with the squad to play in their 2-0 friendly victory over Manchester City on July 13.
Both players impressed against the English champions, with Tankulic scoring and Konrad producing a strong defensive display.