We study memoryless interference channels with gradual data arrival in the absence of feedback. The information bits arrive at the transmitters according to independent and asynchronous~(Tx-Tx asynchrony) Bernoulli processes with average data rate $λ$. Each information source turns off after generating a number of $n$ bits. In a scenario where the transmitters are unaware of the amount of Tx-Tx asynchrony, we say $ε$ is an \textit{achievable outage level} in the asymptote of large~$n$ if (i) the average transmission rate at each transmitter is $λ$ and (ii) the probability that the bit-error-rate at each receiver does not eventually vanish is not larger than~$ε$. Denoting the infimum of all achievable outage levels by $ε(λ)$, the contribution of this paper is an upper bound (achievability result) on $ε(λ)$. The proposed method of communication is a simple block transmission scheme where a transmitter sends a random point-to-point codeword upon availability of enough bits in its buffer. Both receivers that treat interference as noise or decode interference are addressed.