Belgium take on France in St. Petersburg on Tuesday, while the winners of tomorrow's two quarter-finals will meet in Moscow on Wednesday.
"President Putin called me before the (last-16) game with the Spanish and after", Cherchesov said.
"In any case, it definitely won't be a failed World Cup now, and we'll wait to find out our opponents and see what happens".
Instead, the former Arsenal frontman - who moved to Stamford Bridge for about £18m in January - is preferred for his link play with Antoine Griezmann and Kylian Mbappe, who have made most of the headlines with their goals in Russian Federation.
"It was important to gain a good position with respect to these players". "Against Argentina [in the last 16], he made an assist for Mbappe and we need this sort of supporting play".
"We've already had our first knockout win in 12 years, our first win in a penalty shootout in a World Cup for England, the highest number of goals scored in an individual game (against Panama)", added former defender Southgate.
"I work a lot and I'm fully focused on the things that I'm doing now". He had 11 saves and had allowed only one goal in four World Cup matches in Russian Federation.
Deschamps and his side have come in for criticism for their style of play, but after the game he said he was pleased with the way the 1998 champions were playing.
"That's what I told all the players".
"For now, we have the potential to be in the semi-finals, that is certain", he said.
'It's not going to be easy and we're not conceited at all but we don't think it's impossible; it's hard but not impossible.