The Congress chief said that while the doors of his party are open, time is running out, but Kejriwal slammed him, questioning what U-turn was he talking about as the talks were still on.
He also accused Gandhi of helping the BJP by "dividing anti-Modi votes in Uttar Pradesh and in some other states".
Amid this sparring on Twitter, sources claimed the two sides were still "engaged" in taking the alliance talks forward and are likely to meet again on Wednesday over seat sharing between the two.
The Congress chief said his party is willing to give four of the seven Lok Sabha seats in Delhi to the AAP to ensure that their alliance defeats the BJP, which had won all the seven seats in 2014.
"An alliance between the Congress and the AAP in Delhi would mean the rout of the BJP. The Congress is willing to give up four seats in Delhi to the AAP to ensure this.
"But, Mr Kejriwal has done yet another U-turn. Our doors are still open, but the clock is running out," Gandhi tweeted.
He used the hashtag "AbAAPkiBaari" to target the AAP leadership to come out with its side for an alliance with the Congress.
Hitting back, Delhi Chief Minister Kejriwal said, "What U-turn. The talks are still on."
"Your tweet shows that an alliance is not your desire but only a pretension. I am pained that you are giving mere statements," he said.
"Today, the foremost issue is to save the country from the threat of Modi-Shah. Unfortunately, you are helping Modi by dividing anti-Modi votes in Uttar Pradesh and in some other states," Kejriwal tweeted in Hindi.
Congress insiders claimed that party's incharge PC Chacko Monday sent messages to AAP leaders Manish Sisodia and Sanjay Singh, seeking their answer.
"The AAP leaders have been asked to make up their mind on alliance in Delhi alone considering the ground realities," they said.
AAP sources said that Singh and senior Congress leader Ahmad Patel will meet on Wednesday for discussing alliance prospects.
The Congress has offered the AAP 4:3 seat-sharing formula, four Lok Sabha seats for the ruling party and three seats for itself.
The arrangement has been arrived at on basis of the vote share of both parties in 2017 civic polls here, they said.
The Congress has accused the Kejriwal-led outfit of backing out from its commitment, as it is demanding seats in Goa, Haryana, Punjab and Chandigarh, besides Delhi for a pre-poll understanding with the Congress.
AAP leader Sanjay Singh said in response to Gandhi's tweet, "The Congress is not giving even a single seat to the AAP in Punjab where it has four MPs and 20 MLAs, and in Haryana."
"In Delhi where the Congress has zero MPs and MLAs, it is demanding three seats. Is this how agreement happens? Why do not you want to stop the BJP in other states," he asked.
The AAP has already announced its candidates on the seven parliamentary constituencies in Delhi, while the Congress has finalised names of its candidates which could be announced soon.