The potential marriages for Edward II's daughters mentioned in Ralph Basset's letter were with the offspring of the powerful Charles de Valois, count of Valois, Alençon, Perche, Anjou and Maine. Valois (12 March 1270 - 16 December 1325) was the son of Philip III of France and Isabel of Aragon and the brother of Philip IV, and the uncle of Louis X, Philip V, Charles IV and Edward II's queen Isabella. His children and Edward II's therefore were first cousins once removed. His eldest son Philip, born in 1293, succeeded his cousin Charles IV in 1328 as the first of the Valois kings of France, as Charles and his two elder brothers had no surviving sons and therefore were the last of the Capetian kings. Valois was married three times and had about fourteen children, including Jeanne, mother of Edward III's queen Philippa of Hainault; Isabelle, who married the heir of the duke of Brittany; Catherine, titular empress of Constantinople and princess of Taranto; another Isabelle, duchess of Bourbon; Blanche, Holy Roman Empress and queen of Germany; yet another Isabelle, abbess of Fontevrault. Valois was, in the usual tangled manner of royal relationships, also the uncle of Edward II's half-brothers the earls of Norfolk and Kent, being the elder half-brother of their mother Queen Marguerite.
Louis of Chartres died on 2 November 1328, probably aged only ten or thereabouts. John of Valois married Bonne (born Jutta) of Bohemia, daughter of John the Blind, king of Bohemia, in 1332 and succeeded his father as King John II in 1350. He is known to history as Jean le Bon, John the Good, and was succeeded by his son Charles V and then his grandson Charles VI, and so on. No-one could have known at the time of Charles de Valois's marriage proposals in January 1324 that Charles IV would die without a son in February 1328, and that if this proposal had been realised it would ultimately have made one of Edward II's daughters queen of France. Given that Edward's son Edward III claimed the French throne from his kinsman Philip VI (who was also the uncle of Edward's queen, Philippa), that's a fascinating what-if.
The marriage of the future Edward III to one of Charles of Valois's daughters had also been proposed, incidentally, in about May 1323; Edward II told Valois and Charles IV on 6 June that he would put the suggestion to his next parliament, which didn't take place until February 1324, by which time the alternative marriage proposal had been suggested.  The daughter is not named, but presumably meant one of Valois's daughters with Mahaut of St Pol, who were all about the right age to marry Edward of Windsor: Marie (b. c. 1309), Isabelle (b. 1313) or Blanche (b. 1317). In March 1324, Edward II sent envoys to Jaime II of Aragon to discuss a marriage between Edward of Windsor and Jaime's daughter Violante, and in February 1325 sent envoys to Castile to arrange a marriage for the boy with Alfonso XI's sister Leonor. 
The impetus for the Valois marriages came from Charles de Valois himself. The next proposed marriage, however, was one which Ralph Basset had been trying to negotiate on Edward II's behalf. The prospective bridegroom was Edward's half-brother Edmund of Woodstock, earl of Kent, and the prospective bride was Régine de Got, daughter and heir of Bertrand de Got, viscount of Lomagne, a small town north-west of Toulouse (Basset wrote: jeo avoi comence parlaunce et trettement od le viscounte de Leomaine pur avoir euz le mariage de sa file et mon seignur vostre frere le counte de Kente, "I had begun discussions and negotiations with the viscount of Lomagne to have had the marriage of his daughter and my lord your brother the earl of Kent").
Ralph Basset informed Edward in his letter that unfortunately his negotiations with Bertrand had been unsuccessful and he had heard that Régine de Got was shortly to marry the count of Armagnac instead (a ceo qe jeo ay entendu ele serra mariee au counte de Armeniak en moult bref temps). The count of Armagnac in 1324 was Jean I, who was still underage at the time and lived until 1373. Countess Régine sadly did not live long: John Travers, constable of Bordeaux, told Edward II on 1 September 1325 that "the countess of Armagnac, who was the daughter of the viscount of Lomagne, is dead without an heir of her body," and on the 23rd Edward wrote to inform his half-brother the earl of Kent, the spurned bridegroom and the king's lieutenant in Gascony.  Betrand de Got, viscount of Lomagne, died in 1324, within months of Basset's letter. Ralph Basset was afraid of an alliance between the viscount of Lomagne, the Armagnacs and Amanieu, lord of Albret against Edward II, which he had been hoping to avert with the marriage of Kent and Régine. Armagnac and Albret supported Charles IV against Edward II in the War of Saint-Sardos, although Albret's son Bérard was a staunch ally of Edward. Malcolm Vale's 1990 book The Origins of the Hundred Years War: The Angevin Legacy 1250-1340 mentions Kent and Régine's proposed marriage (p. 94 footnote 87), but otherwise I don't believe it's ever been discussed, except here.
1) Pierre Chaplais, ed., The War of Saint-Sardos (1323-1325): Gascon Correspondence and Diplomatic Documents (1954), pp. 15-17.
2) Pierre Chaplais, English Medieval Diplomatic Practice (1982), part 1, vol. 1, pp. 64-66.
3) Foedera 1307-1327, p. 523; Close Rolls 1318-1323, pp. 713-714.
4) Foedera 1307-1327, pp. 548-549; Close Rolls 1323-1327, p. 171; Patent Rolls 1324-1327, pp. 103-104; Chaplais, War of Saint-Sardos, pp. 214-216.
5) Patent Rolls 1307-1313, p. 83; Foedera 1307-1327, p. 51.
6) Gascon Rolls C 61/32, mem. 17.
7) Chaplais, War of Saint-Sardos, p. 240; Foedera 1307-1327, pp. 609-610.