RCB are placed fifth and RR sixth on the points table with two points each from two matches.
They both lost their respective openers against Kolkata Knight Riders and Sunrisers Hyderabad.
RCB notched up their first victory of the season, riding on South African star A B de Villiers' brilliant half-century to beat Kings XI Punjab by four wickets.
RR celebrated their return to the 'fortress' Sawai Mansingh Stadium with a 10-run win over Delhi Daredevils in the rain-hit match, getting past a revised target of 71 runs in six overs as per the Duckworth/Lewis method.
De Villiers and Quinton de Kock, who scored 45 runs to lay the foundation against Punjab, will aim to continue the good work.
Captain Virat Kohli, who has 52 runs under his belt so far, will be itching to make a mark by getting a big knock.
Brendon McCullum also would like to get into the thick of things, especially after getting a golden duck.
Sarfaraz Khan has been disappointing as he managed to get only six runs in two matches and if he does not pulls his socks up quickly, he could lose his place to England's Moeen Ali, who can also bowl a bit.
Chris Woakes, though successful with the ball, is yet to make an impression with his bat.
The RCB think-tank, including Gary Kirsten, could give a chance to Corey Anderson, who has the ability to hit long and hard.
Both Umesh Yadav and Chris Woakes, who have picked up 5 wickets each, would be keen to snatch the purple cap from MI's Mayank Markande, who has bagged 7 seven wickets so far.
Much was expected from the spin duo of Yuzuvendra Chahal and Washington Sundar but they have managed to get only 4 wickets between them.
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The in-form New Zealand quickie Tim Southee might get a chance to swing his arms.
RR captain Ajinkya Rahane and Sanju Samson, who staged a recovery with a 45 off 40 balls and 37 off 22 balls, respectively, would like to continue in similar vein.
Rahane has scored 58 runs and Samson 86, with 49 being his highest score.
Jos Buttler and Rahul Tripathi would want to provide entertainment in the death overs as they did against Delhi.
Buttler had hammered two fours and as many sixes in his entertaining knock before being bowled off a slower-ball from Mohammad Shami.
Big Bash star D'Arcy Short will also look to get going, especially after getting only 10 runs in two games.
RR's bowling attack comprising pacers Ben Laughlin and Dhawal Kulkarni had done well to keep DD batsmen in check with their variations.
Ben Stokes, the most-expensive signing at the IPL Player Auction 2018, hasn't been in the best of batting form recently, but RR will hope he can start hitting a few balls out of the park soon. He has just managed 21 runs in two games.
Royal Challengers Bangalore: Brendon McCullum, Quinton de Kock, Virat Kohli (capt), AB de Villiers, Sarfraz Khan, Mandeep Singh, Chris Woakes, Washington Sundar, Umesh Yadav, Yuzvendra Chahal, Kulwant Khejroliya
Rajasthan Royals: Heinrich Klassen, Benjamin Stokes, Jaydev Unadkat, Sanju Samson, Jofra Archer, Krishnappa Gowtham, Jos Buttler, Ajinkya Rahane, Darcy Short, Rahul Tripathi, Dhawal Kulkarni, Zahir Khan Pakteen, Ben Laughlin, Stuart Binny, Dushmantha Chameera, Anureet Singh, Aryaman Vikram Birla, Midhun S, Shreyas Gopal, Prashant Chopra, Jatin Saxena, Ankit Sharma, Mahipal Lomror.