The duo will likely cost a combined £70 million as Chelsea aim to negotiate Leicester's asking price to below the £40m that they have been demanding for England midfielder Drinkwater.
Meanwhile, Chelsea have no serious competition in the race for the versatile Oxlade-Chamberlain as Liverpool's interest has cooled. But, despite the 24-year-old having less than a year left on his contract at Arsenal, he will likely cost between £35m and £40m.
Oxlade-Chamberlain is keen to leave Arsenal, despite playing in the club's 4-0 defeat at Liverpool on Sunday, and is hoping to earn around £150,000-a-week in his next contract. The former Southampton is not willing to accept any deal to remain at the Emirates Stadium.
Chelsea are keen to bring in English players ahead of the transfer window closing after failing to field one in the past two games. Fikayo Tomori was Chelsea's only English player in their match day squad for Sunday's 2-0 win over Everton and he failed to make an appearance off the bench.
Antonio Conte doesn't conduct his own transfers at Chelsea but he has made made several calls for his club to strengthen as he has been faced with adding non-senior players to his bench in recent weeks.
Conte spoke about bringing in players Sunday's win over Everton, saying: “There are four days to go, no, in the transfer market. I think the club is working very hard to try to improve our squad.
"Don't forget when we start, next month, in September we have to play seven games. I need to rotate my players because it's very difficult to play seven games in a month with some players.
"I think the club is working very hard to strengthen the squad. They know very well our situation. But I repeat, I'm ready in every case to continue to work with this group of players, with the players we have.
“I think my message for the fans is always the same: I'm totally committed to the club. Totally committed to improve my players and my team. I'm a coach, not a manager. I think the best of my work is on the pitch, to try and improve my players and my team.
"Then, for sure, when you want to strengthen your squad, you have to give your opinion and speak with your club, but then the club goes into the transfer market to try and sort the situation. To try and help us. Sometimes it could be possible. Sometimes it is possible. Sometimes it's not possible.
"But I must be focused with things on the pitch and continue to work with my players.”