Since it is not clear whether your question relates to the time before or after enrollment for Hajj, we shall deal in our answer to your question with both the two suppositions:
Saving before the Enrollment
The grand jurists are generally of the view that khums is applicable to the saving. However some other jurists say that "Khums is not payable on the money he spends on purchasing a house which he needs and procuring his necessities and essential things provided that there is no other way to purchase the house and procure the necessities."
The edicts of some of the grand jurists concerning saving are as follows:
1. Grand Ayatollah Khamenei: Khums is payable on the property (money) acquired by way of trade, if it exceeds the annual expenses of a person. In case the saving is used to purchase the essential things and meet the expenses, Khums would not be payable on it, if it is spent (two or three months) shortly after the khums year to meet those needs.
2. Late Ayatollah Fazel Lankarani has said: "It is necessary to pay Khums on saving."
3. Late Grand Ayatollah Bahjat: If a person needs to have his own house, buy household accessories and prepare his daughter's dowry, and he cannot purchase all this at once and his annual income does not suffice for purchasing a house and he has to save money for a couple of years to be able to buy a house, Khums is not payable on them. For example, if he buys the land for the house in the first year and the building materials in the second year, khums would not be payable on the iron, bricks and other materials even though they may have remained unused for sometime. Even if he may save money and in case he cannot purchase all those things with that money at once and he can buy them in a few years, say two years or three, and in the eye of the public he is considered to be in need of that thing and the money is saved for purchasing that thing, again Khums will not be payable on the saving. In case, however, he purchases that thing after a long time, for example after twenty years, in a way such that the custom will not say that he is in need of it, in this case, khums would be payable on the money which he has saved and over which a year has elapsed.
As for whether or not Khums is applicable to the money which is deposited into Hajj account, the details shall follow:
Question: Is Khums payable on the money which is deposited into bank account for registration and enrollment for Hajj keeping in view that the money is from the income of the same year (in which the money was deposited) and over which a year has passed now?
Late Imam Khomeini and Ayatollah Fazel Lankarani: If Hajj had been obligatory on him and going to Hajj had not been possible except through enrolling himself and making the deposit, Khums is not payable, or else, Khums should be paid from it.
Grand Ayatollah Khamenei: It is not obligatory to pay Khums on the money which has been paid for enrollment for Hajj and Umrah, if it has been paid as the expenses for the hajj journey according to the contract concluded between the individual and Hajj and Pilgrimage Organization. In case, the money is taken back, Khums would be payable on it.
Bahjat, Makarem and Saafi: Khums is not payable on the money.
Sistani: If going to Hajj is not possible through other ways (without going through hardship), Khums is not payable.
Tabrizi and Wahid Khurasani: Khums is payable on the money.
Noori: If he has earned the money in the same year in which he became financially capable of performing Hajj, Khums would not be payable on it.
 - Index No.696 (website 744).
 - Jami'ul Masail, Grand Ayatollah Fazel Lankarani, vol.1, pg.171, question 654, Mehr Publication, Second edition, 1996.
 - Tauzihul Masail, Grand Ayatollah Muhammad Taqi Bahjat, pg.272, Issue No.1391, Amir Publications, Qom, 18th edition, 1999.
 - Imam Khomeini, Istifta'at, vol.1, q.43; Fazel Lankarani, Jami'ul Masail, vol.2, q.493.
 - Ibid, pg.97
 - Ahkam-e Khums, Sayed Mujtaba Husseini, pg.63, Ma'aref Publications Office, fourth edition, 2006.
 - Office of Grand Ayatollah Sistani
 - Tabrizi, Istifta'at, question 873; Office of Ayatollah Wahid Khurasani
 - Noori, Istifta'at, vol.2, pg.324.