Kante missed the Blues' 3-2 win over Norwich City on Saturday with an ankle injury.
Per Goal's Nizaar Kinsella, Lampard said:
"It is a difficult one. He has had four years of constantly playing. [The] injury from the Europa League final has affected his pre-season. Then he hurt himself against Man United.
"We are trying to get him through it. Today was too painful. The international break will be great for him to get the injury right. It would be great for everybody [if Kante stays] because he's been playing game after game after game."
Kante was expected to miss last season's UEFA Europa League final against Arsenal with a knee injury, but he started the showpiece thanks to an injection and put in a fine performance as Chelsea won 4-1.
He completed 120 minutes in the UEFA Super Cup against Liverpool, though, and impressed while doing so (U.S. only):
Liverpool.com's Joel Rabinowitz singled him out for praise:
The midfielder is known for his remarkable energy and endurance, and despite his exertions in the Super Cup, those qualities were on show again soon after in Chelsea's 1-1 draw with his former club Leicester:
Kante is among the most influential players in the Premier League, so he's naturally a key player at Stamford Bridge.
Keeping him fit is paramount to their ambitions, so it would be of enormous benefit for him not to be involved in France's UEFA Euro 2020 qualifiers against Albania on September 7 and Andorra on September 10.
He's similarly important for Les Bleus, but they should be capable of winning those matches without him. They'll also benefit long term if the break allows him to get back to peak condition, so letting him remain at Stamford Bridge would be best for all parties.