Bertrand, a Champions League winner with Chelsea who joined Southampton in 2015, is attracting interest from City having been linked with a move to the Etihad last summer.
Northern Ireland international Evans is also on City’s radar, according to the Daily Mail, but Guardiola will have to see off competition from Arsenal to sign the former Manchester United man.
City are also believed to be tracking Real Sociedad’s Inigo Martinez, Espanyol left back Aaron Martin and 24-year-old Augsburg star Philipp Max.
Guardiola’s City maintained their incredible run this season with a 1-0 victory over Newcastle United at St James’ Park on Wednesday evening.
England international Raheem Sterling scored the only goal of the game in the first half and Guardiola said after the win: ‘We did absolutely everything but it is difficult to play when the other team doesn’t want to play.
‘In the last minutes we played in their rhythm and then it was not easy because it is not over at 1-0 – we created enough chances to win 2-0, 3-0, 4-0.
‘As a manager I have to adapt. We have played teams who high press, low press, attack us. Any manager and team can play how they want. And you have to find a way to beat them.’
The victory was City’s 11th consecutive away from home, matching the Premier League record set by Chelsea in 2008. The league leaders face Crystal Palace on Sunday.