It is permissible to use contact lenses when fasting, and the fact that they are first placed in a cleansing solution does not matter, because the worst that can happen is that some of this solution leaks into the eye. Thus it is like eye drops, and eye drops do not invalidate the fast, according to the correct scholarly opinion. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: There is nothing wrong with a fasting person using kohl, or putting drops in his eyes, or putting drops in his ears; even if he notices the taste of that in his throat, that does not invalidate his fast, because it is not food or drink, and is not like food or drink. The evidence forbids eating and drinking, and that does not include things that are not like food or drink. What we have mentioned is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) and it is the correct view. End quote. 

Fiqh al-‘Ibaadaat, p. 191 

See also questions no. 2299, 22199, 38023 

And Allaah knows best.