Clubs can not draw teams from the same country and will also be separated into seeds for the first round, decided by UEFA's co-efficient rankings established at the start of the season.
Rangers are among the teams they could face in the first qualifying round. Champions League play-off losers automatically make it to the Europa League group stage like Skenderbeu did in the 2015-16 campaign.
The Europa League's first and second qualifying rounds will also be drawn on Monday and 11 British sides will be pulled out of the pot.
Welsh Cup winners Bala Town were drawn against Vaduz of Liechtenstein while Derry City will play Danish outfit Midtjylland.
The Europa League's first round ties will take place on June 29 and July 6.
Should TNS clear the first round of qualifying, they will then face HNK Rijeka, who finished top of the Croatian HNL table - two points ahead of Dinamo Zagreb.
League of Ireland side Derry City will face FC Midtjylland (Denmark), who beat Manchester United a year ago, while Shamrock Rovers take on Stjarnan (Iceland) and Cork City come up against Levadia Tallinn (Estonia).