There are a few narratives that show our prophet's prophecy on this battle.
A report showing that 'Ali was informed
One report which is close to what you've asked about is the one that @UmH already mentioned:
Here and in the following I'll be translating from Arabic language as these translations are of my own take them carefully
26657 حَدَّثَنَا حُسَيْنُ بْنُ مُحَمَّدٍ قَالَ حَدَّثَنَا الْفُضَيْلُ يَعْنِي ابْنَ سُلَيْمَانَ قَالَ حَدَّثَنَا مُحَمَّدُ بْنُ أَبِي يَحْيَى عَنْ أَبِي أَسْمَاءَ مَوْلَى بَنِي جَعْفَرٍ عَنْ أَبِي رَافِعٍ أَنَّ رَسُولَ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ قَالَ لِعَلِيِّ بْنِ أَبِي طَالِبٍ إِنَّهُ سَيَكُونُ بَيْنَكَ وَبَيْنَ عَائِشَةَ أَمْرٌ قَالَ أَنَا يَا رَسُولَ اللَّهِ قَالَ نَعَمْ قَالَ أَنَا قَالَ نَعَمْ قَالَ فَأَنَا أَشْقَاهُمْ يَا رَسُولَ اللَّهِ قَالَ لَا وَلَكِنْ إِذَا كَانَ ذَلِكَ فَارْدُدْهَا إِلَى مَأْمَنِهَا
Hussain ibn Muhammad informed us (and) said al-Fudail -means ibn Sulaiman- said we were inforemed by Muhammad ibn abi Yahya from abu Asma' mawala bani Ja'afar from abi Rafi'i that the Messenger of Allah Allah's prayers and blessings be upon him said to 'Ali ibn abi Talib that there will be a matter between you and 'Aisha. He said, me O Messenger of Allah? He said: Yes, he said: Yes. He said: (then) I'm the most miserable one, O Messenger of Allah. He said no, but if so, return her place of safety.
(Source)
As it is stated the in the answer above this hadith can be found in Musnad Ahmad, at-Tabarani's al-Mo'jam al-Kabir and al-Bzaar's Musnad. Ibn Haitham -in his Majma' az-Zawaaid- claimed that it's narrators are trustworthy.
However both ibn Hajar and Yahya ibn Ma'iyn consider al-Fudail not to be trustworthy or at least not more than sadooq with many mistakes. This hadith also appears in ibn Jawzi's al-'ilal al-Mutanahiya, in at-Tahawi's Mushkil al-Athar and in Kanz al-'Ummal of al-Mutaqi al-Hindi who marked it as da'if .
Note that this narrative only shows that the prophet () informed 'Ali() not 'Aishah().
One report in which both 'Ali() and 'Aishah() were informed at once
In fact there's a hadith on the authorithy of Umm Salamah () that seems to fit your inquiry in al-Hakim (from Nishapur's) al-Mustadrak المستدرك على الصحيحين -see here in Arabic-:
4668 - حدثنا أبو بكر محمد بن عبد الله الحفيد ، ثنا أحمد بن نصر ، ثنا أبو نعيم الفضل بن دكين ، ثنا عبد الجبار بن الورد ، عن عمار الدهني ، عن سالم بن أبي الجعد ، عن أم سلمة - رضي الله عنها - قالت :
ذكر النبي - صلى الله عليه وآله وسلم - خروج بعض أمهات المؤمنين ، فضحكت عائشة ، فقال : " انظري يا حميراء ، أن لا تكوني أنت " ثم التفت إلى علي فقال : " إن وليت من أمرها شيئا فارفق بها .
We were informed by abu Bakr ibn Muhammad ibn 'Abdullah al-Hafeed from Ahmad ibn Nasr from abu Nu'aym al_Fadl ibn Dukayn from 'Abdul-Jabbar ibn al-Ward from 'Ammar ad-Duhny from Salim ibn abi al-Ja'ad from Umm-Salamah may Allah be pleased with her who said:
The Prophet Allah’s prayers and blessings be upon him and his family - mentioned the exodus of some of the mothers of the believers, and Aisha laughed, and he said: “Look, Humayra', that it should not be you.” Then he turned to 'Ali and said: “If she was entrusted to you, then be gentle to her.
Al-Hakim claimed that this hadith is in accordance to the condition of both Sahihs, but a-Dhahabi pointed out that none of them ever reported via 'Abdul-Jabbar ibn al-Ward, whom basically is not a very weak narrator. A contemporary scholar called Sheikh Sa'ad al-Humaid qualified this hadith as da'if.
A report showing that 'Aishah() was warned
A third report which appears in Musnad Ahmad (see here -this is the version of the hadith displayed below), in Sahih ibn Hebban (See here), in al-Mustadrak (of al-Hakim from Nishapur -see here), in Mossanf ibn abi Shabyah (see (15) here):
(24254) - (6\52) عن إسماعيل، حدثنا قيس قال: لما أقبلت عائشة بلغت مياه بني عامر ليلا نبحت الكلاب، قالت: أي ماء هذا؟ قالوا: ماء الحوأب قالت: ما أظنني إلا أني راجعة فقال بعض من كان معها: بل تقدمين فيراك المسلمون، فيصلح الله عز وجل ذات بينهم، قالت: إن رسول الله صلى الله عليه وسلم قال لنا ذات يوم: " كيف بإحداكن تنبح عليها كلاب الحوأب؟".
From Isma'il we were informed by Qais who said: When Aishah reached the waters of Bani 'Amer at night, the dogs barked. She said: What water is this? They said: The water of al-Haw'ab (A location between Mekka and Basra)! She said: I do not think (I'll move on) but I am returning back, so some of those who were with her said: Rather advance, and the Muslims will see you, so Allah the Almighty will bless a soul among them. She said: The Messenger of Allah, Allah’s prayers and blessings be upon him, told us one day: “How is it that one of you the dogs al-Haw'ab barking on her?”
Some helpful sources in my investigation:
This article -in Arabic- on some reports and false claims about 'Aishah () on alukah.net.
Fatwa islamqa #240359 -in Arabic-.
