The Grand Alliance disintegrated last year after Nitish Kumar walked out of the grouping over graft charge on his then deputy Tejaswi Yadav. He later joined hands with the BJP to form a coalition government in the state in July 2017.
The RJD and the Congress are still continuing together as rivals of ruling JDU and BJP.
Signalling a tiff, Congress Legislature Party leader Sadanand Singh told reporters, "I would urge the party high command to take a decision in favour of fielding our own candidate from Bhabhua even if the RJD insists on contesting the assembly seat".
Mr Singh was responding to a query on reports about jailed RJD supremo Lalu Prasad having said in Ranchi that his party was thinking of contesting "all the seats" where bypolls were scheduled next month.
Former BPCC president Ashok Choudhary said, "We must contest the Araria parliamentary constituency which has a sizeable population of the minorities. We are a national party and a victory from such a seat will help us build a momentum in our favour."
Bihar will see bypolls on March 11 on the Araria Lok Sabha seat along with that of two assembly constituencies, Jehanabad and Bhabhua, necessitated by death of the incumbents.
While Araria and Jahanabad were held by the RJD, Bhabhua was with the BJP.
The Congress had contested Bhabhua the seat in 2015 assembly polls, which it fought in alliance with the RJD and the JD(U).
Acting BPCC president Kaukab Qadri, however, said, "Neither we nor the RJD has taken any official decision on the by-polls so far. I am sure we will reach an agreement that is mutually beneficial. Our top priority is to ensure emphatic victory for the Mahagathbandhan in all the three seats."
He also said, "As a matter of fact, there is no dispute over the Araria (Lok Sabha seat) and Jahanabad assembly seat.
Both were represented by the RJD and we respect our ally's claim to these.
"In Bhabhua, our party has a strong base. That is why we hold the view that the Congress should be allowed to contest the seat. But, I don;t think it would involve any confrontation with the RJD. Leaders of both the parties will sit together and arrive at a solution," Mr Qadri said.
Asked to comment on the statements of Mr Singh and Mr Choudhary, RJD vice president and former chief minister Rabri Devi told reporters, "If the Congress wants to contest any seat, its leaders should come and meet us. What would issuing statements in the media achieve".
Talking to PTI over phone from Ranchi, RJD national general secretary Bhola Yadav said, "The RJD is on a very strong footing in Bhabhua and the seat has been won by our party or our allies a number of times".
"The Congress has been in a bad shape over the years.
Yet, if it insists on contesting the seat, leaders of the two parties can sit together and arrive at an amicable solution."
The RJD lawmaker said his party's parliamentary board today met at Patna and authorised Lalu Prasad to take a call on all the three seats where by-polls are due.
The NDA is also yet to arrive at a decision on fielding candidates on the seats. Nitish Kumar, however, yesterday made it clear the JD(U) would not contest any of the three.
The NDA choice of candidates would from among the BJP, the RLSP and the HAM of Jitan Ram Manjhi.